Finite word¶
AUTHORS:
- Arnaud Bergeron
- Amy Glen
- Sébastien Labbé
- Franco Saliola
- Julien Leroy (March 2010): reduced_rauzy_graph
EXAMPLES:
Creation of a finite word¶
Finite words from python strings, lists and tuples:
sage: Word("abbabaab")
word: abbabaab
sage: Word([0, 1, 1, 0, 1, 0, 0, 1])
word: 01101001
sage: Word( ('a', 0, 5, 7, 'b', 9, 8) )
word: a057b98
Finite words from functions:
sage: f = lambda n : n%3
sage: Word(f, length=13)
word: 0120120120120
Finite words from iterators:
sage: from itertools import count
sage: Word(count(), length=10)
word: 0123456789
sage: Word( iter('abbccdef') )
word: abbccdef
Finite words from words via concatenation:
sage: u = Word("abcccabba")
sage: v = Word([0, 4, 8, 8, 3])
sage: u * v
word: abcccabba04883
sage: v * u
word: 04883abcccabba
sage: u + v
word: abcccabba04883
sage: u^3 * v^(8/5)
word: abcccabbaabcccabbaabcccabba04883048
Finite words from infinite words:
sage: vv = v^Infinity
sage: vv[10000:10015]
word: 048830488304883
Finite words in a specific combinatorial class:
sage: W = Words("ab")
sage: W
Words over {'a', 'b'}
sage: W("abbabaab")
word: abbabaab
sage: W(["a","b","b","a","b","a","a","b"])
word: abbabaab
sage: W( iter('ababab') )
word: ababab
Finite word as the image under a morphism:
sage: m = WordMorphism({0:[4,4,5,0],5:[0,5,5],4:[4,0,0,0]})
sage: m(0)
word: 4450
sage: m(0, order=2)
word: 400040000554450
sage: m(0, order=3)
word: 4000445044504450400044504450445044500550...
Functions and algorithms¶
There are more than 100 functions defined on a finite word. Here are some of them:
sage: w = Word('abaabbba'); w
word: abaabbba
sage: w.is_palindrome()
False
sage: w.is_lyndon()
False
sage: w.number_of_factors()
28
sage: w.critical_exponent()
3
sage: print w.lyndon_factorization()
(ab, aabbb, a)
sage: print w.crochemore_factorization()
(a, b, a, ab, bb, a)
sage: st = w.suffix_tree()
sage: st
Implicit Suffix Tree of the word: abaabbba
sage: st.show(word_labels=True)
sage: T = words.FibonacciWord('ab')
sage: T.longest_common_prefix(Word('abaabababbbbbb'))
word: abaababa
As matrix and many other sage objects, words have a parent:
sage: u = Word('xyxxyxyyy')
sage: u.parent()
Words
sage: v = Word('xyxxyxyyy', alphabet='xy')
sage: v.parent()
Words over {'x', 'y'}
Factors and Rauzy Graphs¶
Enumeration of factors, the successive values returned by next(it) can appear in a different order depending on hardware. Therefore we mark the three first results of the test random. The important test is that the iteration stops properly on the fourth call:
sage: w = Word([4,5,6])^7
sage: it = w.factor_iterator(4)
sage: next(it) # random
word: 6456
sage: next(it) # random
word: 5645
sage: next(it) # random
word: 4564
sage: next(it)
Traceback (most recent call last):
...
StopIteration
The set of factors:
sage: sorted(w.factor_set(3))
[word: 456, word: 564, word: 645]
sage: sorted(w.factor_set(4))
[word: 4564, word: 5645, word: 6456]
sage: w.factor_set().cardinality()
61
Rauzy graphs:
sage: f = words.FibonacciWord()[:30]
sage: f.rauzy_graph(4)
Looped digraph on 5 vertices
sage: f.reduced_rauzy_graph(4)
Looped multi-digraph on 2 vertices
Left-special and bispecial factors:
sage: f.number_of_left_special_factors(7)
1
sage: f.bispecial_factors()
[word: , word: 0, word: 010, word: 010010, word: 01001010010]
-
class
sage.combinat.words.finite_word.
CallableFromListOfWords
¶ Bases:
tuple
A class to create a callable from a list of words. The concatenation of a list of words is obtained by creating a word from this callable.
-
class
sage.combinat.words.finite_word.
Factorization
¶ Bases:
list
A list subclass having a nicer representation for factorization of words.
TESTS:
sage: f = sage.combinat.words.finite_word.Factorization() sage: f == loads(dumps(f)) True
-
class
sage.combinat.words.finite_word.
FiniteWord_class
¶ Bases:
sage.combinat.words.abstract_word.Word_class
-
BWT
()¶ Returns the Burrows-Wheeler Transform (BWT) of self.
The Burrows-Wheeler transform of a finite word
is obtained from
by first listing the conjugates of
in lexicographic order and then concatenating the final letters of the conjugates in this order. See [1].
EXAMPLES:
sage: Word('abaccaaba').BWT() word: cbaabaaca sage: Word('abaab').BWT() word: bbaaa sage: Word('bbabbaca').BWT() word: cbbbbaaa sage: Word('aabaab').BWT() word: bbaaaa sage: Word().BWT() word: sage: Word('a').BWT() word: a
REFERENCES:
- [1] M. Burrows, D.J. Wheeler, “A block-sorting lossless data compression algorithm”, HP Lab Technical Report, 1994, available at http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-124.html
-
abelian_complexity
(n)¶ Return the number of abelian vectors of factors of length
n
of self.EXAMPLES:
sage: w = words.FibonacciWord()[:100] sage: [w.abelian_complexity(i) for i in range(20)] [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
sage: w = words.ThueMorseWord()[:100] sage: [w.abelian_complexity(i) for i in range(20)] [1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2]
-
abelian_vector
(alphabet=None)¶ Return the abelian vector of self counting the occurrences of each letter.
The vector is defined w.r.t the order of the alphabet of the parent. See also
evaluation_dict()
.INPUT:
self
– word having a parent on a finite alphabetalphabet
– DEPRECATED
OUTPUT:
listEXAMPLES:
sage: W = Words('ab') sage: W('aaabbbbb').abelian_vector() [3, 5] sage: W('a').abelian_vector() [1, 0] sage: W().abelian_vector() [0, 0]
The argument alphabet is deprecated:
sage: Word('aabaa').abelian_vector('abc') doctest:...: DeprecationWarning: The argument alphabet of methods abelian_vector and parikh_vector is deprecated and will be removed in a future version of Sage. In order to fix this, you must define your word on a parent with a finite alphabet. See http://trac.sagemath.org/17058 for details. [4, 1, 0]
You may fix the above deprecated use of alphabet argument this way:
sage: W = Words('abc') sage: W('aabaa').abelian_vector() [4, 1, 0]
TESTS:
sage: W = Words() sage: W('aabaa').abelian_vector() Traceback (most recent call last): ... TypeError: The alphabet of the parent is infinite; define the word with a parent on a finite alphabet or use evaluation_dict() instead
-
abelian_vectors
(n)¶ Return the abelian vectors of factors of length
n
of self.The vectors are defined w.r.t the order of the alphabet of the parent.
OUTPUT:
Set of tuplesEXAMPLES:
sage: W = Words([0,1,2]) sage: w = W([0,1,1,0,1,2,0,2,0,2]) sage: w.abelian_vectors(3) {(1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1)} sage: w[:5].abelian_vectors(3) {(1, 2, 0)} sage: w[5:].abelian_vectors(3) {(1, 0, 2), (2, 0, 1)}
sage: w = words.FibonacciWord()[:100] sage: sorted(w.abelian_vectors(0)) [(0, 0)] sage: sorted(w.abelian_vectors(1)) [(0, 1), (1, 0)] sage: sorted(w.abelian_vectors(7)) [(4, 3), (5, 2)]
The word must be defined with a parent on a finite alphabet:
sage: from itertools import count sage: w = Word(count(), alphabet=NN) sage: w[:2].abelian_vectors(2) Traceback (most recent call last): ... TypeError: The alphabet of the parent is infinite; define the word with a parent on a finite alphabet
TESTS:
sage: W = Words([0, 1]) sage: w = W([0,0,0]) sage: sorted(w.abelian_vectors(3)) [(3, 0)] sage: w = W([0,0,0,1]) sage: sorted(w.abelian_vectors(3)) [(2, 1), (3, 0)] sage: w = W([0,0,0,1,1]) sage: sorted(w.abelian_vectors(3)) [(1, 2), (2, 1), (3, 0)] sage: w = W([0,0,0,1,1,1]) sage: sorted(w.abelian_vectors(3)) [(0, 3), (1, 2), (2, 1), (3, 0)]
sage: w = Word([0,1,0], alphabet=[0,1]) sage: w.abelian_complexity(3) 1 sage: w.abelian_complexity(4) 0
-
apply_permutation_to_letters
(permutation)¶ Return the word obtained by applying permutation to the letters of the alphabet of self.
EXAMPLES:
sage: w = Words('abcd')('abcd') sage: p = [2,1,4,3] sage: w.apply_permutation_to_letters(p) word: badc sage: u = Words('dabc')('abcd') sage: u.apply_permutation_to_letters(p) word: dcba sage: w.apply_permutation_to_letters(Permutation(p)) word: badc sage: w.apply_permutation_to_letters(PermutationGroupElement(p)) word: badc
-
apply_permutation_to_positions
(permutation)¶ Return the word obtained by permuting the positions of the letters in self.
EXAMPLES:
sage: w = Words('abcd')('abcd') sage: w.apply_permutation_to_positions([2,1,4,3]) word: badc sage: u = Words('dabc')('abcd') sage: u.apply_permutation_to_positions([2,1,4,3]) word: badc sage: w.apply_permutation_to_positions(Permutation([2,1,4,3])) word: badc sage: w.apply_permutation_to_positions(PermutationGroupElement([2,1,4,3])) word: badc sage: Word([1,2,3,4]).apply_permutation_to_positions([3,4,2,1]) word: 3421
-
balance
()¶ Returns the balance of self.
The balance of a word is the smallest number
such that self is
-balanced [1].
A finite or infinite word
is said to be
-balanced if for any two factors
,
of
of the same length, the difference between the number of
‘s in each of
and
is at most
for all letters
in the alphabet of
. A
-balanced word is simply said to be balanced. See Chapter 2 of [2].
OUTPUT:
integer
EXAMPLES:
sage: Word('1111111').balance() 0 sage: Word('001010101011').balance() 2 sage: Word('0101010101').balance() 1
sage: w = Word('11112222') sage: w.is_balanced(2) False sage: w.is_balanced(3) False sage: w.is_balanced(4) True sage: w.is_balanced(5) True sage: w.balance() 4
TESTS:
sage: Word('1111122222').balance() 5 sage: Word('').balance() 0 sage: Word('1').balance() 0 sage: Word('12').balance() 1 sage: Word('1112').balance() 1
REFERENCES:
- [1] I. Fagnot, L. Vuillon, Generalized balances in Sturmian words, Discrete Applied Mathematics 121 (2002), 83–101.
- [2] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
-
bispecial_factors
(n=None)¶ Returns the bispecial factors (of length n).
A factor
of a word
is bispecial if it is right special and left special.
INPUT:
n
- integer (optional, default: None). If None, it returns all bispecial factors.
OUTPUT:
A list of words.
EXAMPLES:
sage: w = words.FibonacciWord()[:30] sage: w.bispecial_factors() [word: , word: 0, word: 010, word: 010010, word: 01001010010]
sage: w = words.ThueMorseWord()[:30] sage: for i in range(10): print i, sorted(w.bispecial_factors(i)) 0 [word: ] 1 [word: 0, word: 1] 2 [word: 01, word: 10] 3 [word: 010, word: 101] 4 [word: 0110, word: 1001] 5 [] 6 [word: 011001, word: 100110] 7 [] 8 [word: 10010110] 9 []
-
bispecial_factors_iterator
(n=None)¶ Returns an iterator over the bispecial factors (of length n).
A factor
of a word
is bispecial if it is right special and left special.
INPUT:
n
- integer (optional, default: None). If None, it returns an iterator over all bispecial factors.
EXAMPLES:
sage: w = words.ThueMorseWord()[:30] sage: for i in range(10): ... for u in sorted(w.bispecial_factors_iterator(i)): ... print i,u 0 1 0 1 1 2 01 2 10 3 010 3 101 4 0110 4 1001 6 011001 6 100110 8 10010110
sage: key = lambda u : (len(u), u) sage: for u in sorted(w.bispecial_factors_iterator(), key=key): u word: word: 0 word: 1 word: 01 word: 10 word: 010 word: 101 word: 0110 word: 1001 word: 011001 word: 100110 word: 10010110
-
border
()¶ Returns the longest word that is both a proper prefix and a proper suffix of self.
EXAMPLES:
sage: Word('121212').border() word: 1212 sage: Word('12321').border() word: 1 sage: Word().border() is None True
-
charge
(check=True)¶ Returns the charge of self. This is defined as follows.
If
is a permutation of length
, (in other words, the evaluation of
is
), the statistic charge(
) is given by
where
and
is defined recursively by setting
equal to
if
appears to the right of
in
and
otherwise. Then we set
.
EXAMPLES:
sage: Word([1, 2, 3]).charge() 3 sage: Word([3, 5, 1, 4, 2]).charge() == 0 + 1 + 1 + 2 + 2 True
If
is not a permutation, but the evaluation of
is a partition, the charge of
is defined to be the sum of its charge subwords (each of which will be a permutation). The first charge subword is found by starting at the end of
and moving left until the first
is found. This is marked, and we continue to move to the left until the first
is found, wrapping around from the beginning of the word back to the end, if necessary. We mark this
, and continue on until we have marked the largest letter in
. The marked letters, with relative order preserved, form the first charge subword of
. This subword is removed, and the next charge subword is found in the same manner from the remaining letters. In the following example,
are the charge subwords of
.
EXAMPLE:
sage: w = Word([5,2,3,4,4,1,1,1,2,2,3]) sage: w1 = Word([5, 2, 4, 1, 3]) sage: w2 = Word([3, 4, 1, 2]) sage: w3 = Word([1, 2]) sage: w.charge() == w1.charge() + w2.charge() + w3.charge() True
Finally, if
does not have partition content, we apply the Lascoux-Schutzenberger standardization operators
in such a manner as to obtain a word with partition content. (The word we obtain is independent of the choice of operators.) The charge is then defined to be the charge of this word:
sage: Word([3,3,2,1,1]).charge() 0 sage: Word([1,2,3,1,2]).charge() 2
Note that this differs from the definition of charge given in Macdonald’s book. The difference amounts to a choice of reading a word from left-to-right or right-to-left. The choice in Sage was made to agree with the definition of a reading word of a tableau in Sage, and seems to be the more common convention in the literature.
REFERENCES:
[1] Ian Macdonald, Symmetric Functions and Hall Polynomials second edition, 1995, Oxford University Press
[2] A. Lascoux, L. Lapointe, and J. Morse. Tableau atoms and a new Macdonald positivity conjecture. Duke Math Journal, 116 (1), 2003. Available at: [http://arxiv.org/abs/math/0008073]
[3] A. Lascoux, B. Leclerc, and J.Y. Thibon. The Plactic Monoid. Survey article available at [http://www-igm.univ-mlv.fr/~jyt/ARTICLES/plactic.ps]
TESTS:
sage: Word([1,1,2,2,3]).charge() 4 sage: Word([3,1,1,2,2]).charge() 3 sage: Word([2,1,1,2,3]).charge() 2 sage: Word([2,1,1,3,2]).charge() 2 sage: Word([3,2,1,1,2]).charge() 1 sage: Word([2,2,1,1,3]).charge() 1 sage: Word([3,2,2,1,1]).charge() 0 sage: Word([]).charge() 0
-
cocharge
()¶ Returns the cocharge of self. For a word
, this can be defined as
, where
is the charge of
and
is the evaluation of
, and
is
.
EXAMPLES:
sage: Word([1,2,3]).cocharge() 0 sage: Word([3,2,1]).cocharge() 3 sage: Word([1,1,2]).cocharge() 0 sage: Word([2,1,2]).cocharge() 1
TESTS:
sage: Word([]).cocharge() 0
-
coerce
(other)¶ Tries to return a pair of words with a common parent; raises an exception if this is not possible.
This function begins by checking if both words have the same parent. If this is the case, then no work is done and both words are returned as-is.
Otherwise it will attempt to convert other to the domain of self. If that fails, it will attempt to convert self to the domain of other. If both attempts fail, it raises a TypeError to signal failure.
EXAMPLES:
sage: W1 = Words('abc'); W2 = Words('ab') sage: w1 = W1('abc'); w2 = W2('abba'); w3 = W1('baab') sage: w1.parent() is w2.parent() False sage: a, b = w1.coerce(w2) sage: a.parent() is b.parent() True sage: w1.parent() is w2.parent() False
-
colored_vector
(x=0, y=0, width='default', height=1, cmap='hsv', thickness=1, label=None)¶ Returns a vector (Graphics object) illustrating self. Each letter is represented by a coloured rectangle.
If the parent of self is a class of words over a finite alphabet, then each letter in the alphabet is assigned a unique colour, and this colour will be the same every time this method is called. This is especially useful when plotting and comparing words defined on the same alphabet.
If the alphabet is infinite, then the letters appearing in the word are used as the alphabet.
INPUT:
x
- (default: 0) bottom left x-coordinate of the vectory
- (default: 0) bottom left y-coordinate of the vectorwidth
- (default: ‘default’) width of the vector. By default, the width is the length of self.height
- (default: 1) height of the vectorthickness
- (default: 1) thickness of the contourcmap
- (default: ‘hsv’) color map; for available color map namestype:
import matplotlib.cm; matplotlib.cm.datad.keys()
label
- str (default: None) a label to add on the colored vector.
OUTPUT:
GraphicsEXAMPLES:
sage: Word(range(20)).colored_vector() Graphics object consisting of 21 graphics primitives sage: Word(range(100)).colored_vector(0,0,10,1) Graphics object consisting of 101 graphics primitives sage: Words(range(100))(range(10)).colored_vector() Graphics object consisting of 11 graphics primitives sage: w = Word('abbabaab') sage: w.colored_vector() Graphics object consisting of 9 graphics primitives sage: w.colored_vector(cmap='autumn') Graphics object consisting of 9 graphics primitives sage: Word(range(20)).colored_vector(label='Rainbow') Graphics object consisting of 23 graphics primitives
When two words are defined under the same parent, same letters are mapped to same colors:
sage: W = Words(range(20)) sage: w = W(range(20)) sage: y = W(range(10,20)) sage: y.colored_vector(y=1, x=10) + w.colored_vector() Graphics object consisting of 32 graphics primitives
TESTS:
The empty word:
sage: Word().colored_vector() Graphics object consisting of 1 graphics primitive sage: Word().colored_vector(label='empty') Graphics object consisting of 3 graphics primitives
Unknown cmap:
sage: Word(range(100)).colored_vector(cmap='jolies') Traceback (most recent call last): ... RuntimeError: Color map jolies not known sage: Word(range(100)).colored_vector(cmap='__doc__') Traceback (most recent call last): ... RuntimeError: Color map __doc__ not known
-
commutes_with
(other)¶ Returns True if self commutes with other, and False otherwise.
EXAMPLES:
sage: Word('12').commutes_with(Word('12')) True sage: Word('12').commutes_with(Word('11')) False sage: Word().commutes_with(Word('21')) True
-
complete_return_words
(fact)¶ Returns the set of complete return words of fact in self.
This is the set of all factors starting by the given factor and ending just after the next occurrence of this factor. See for instance [1].
INPUT:
fact
- a non empty finite word
OUTPUT:
Python set of finite words
EXAMPLES:
sage: s = Word('21331233213231').complete_return_words(Word('2')) sage: sorted(s) [word: 2132, word: 213312, word: 2332] sage: Word('').complete_return_words(Word('213')) set() sage: Word('121212').complete_return_words(Word('1212')) {word: 121212}
REFERENCES:
- [1] J. Justin, L. Vuillon, Return words in Sturmian and episturmian words, Theor. Inform. Appl. 34 (2000) 343–356.
-
concatenate
(other)¶ Returns the concatenation of self and other.
INPUT:
other
- a word over the same alphabet as self
EXAMPLES:
Concatenation may be made using
+
or*
operations:sage: w = Word('abadafd') sage: y = Word([5,3,5,8,7]) sage: w * y word: abadafd53587 sage: w + y word: abadafd53587 sage: w.concatenate(y) word: abadafd53587
Both words must be defined over the same alphabet:
sage: z = Word('12223', alphabet = '123') sage: z + y Traceback (most recent call last): ... ValueError: 5 not in alphabet!
Eventually, it should work:
sage: z = Word('12223', alphabet = '123') sage: z + y #todo: not implemented word: 1222353587
TESTS:
The empty word is not considered by concatenation:
sage: type(Word([]) * Word('abcd')) <class 'sage.combinat.words.word.FiniteWord_str'> sage: type(Word('abcd') * Word()) <class 'sage.combinat.words.word.FiniteWord_str'> sage: type(Word('abcd') * Word([])) <class 'sage.combinat.words.word.FiniteWord_str'> sage: type(Word('abcd') * Word(())) <class 'sage.combinat.words.word.FiniteWord_str'> sage: type(Word([1,2,3]) * Word('')) <class 'sage.combinat.words.word.FiniteWord_list'>
-
conjugate
(pos)¶ Returns the conjugate at pos of self.
pos can be any integer, the distance used is the modulo by the length of self.
EXAMPLES:
sage: Word('12112').conjugate(1) word: 21121 sage: Word().conjugate(2) word: sage: Word('12112').conjugate(8) word: 12121 sage: Word('12112').conjugate(-1) word: 21211
-
conjugate_position
(other)¶ Returns the position where self is conjugate with other. Returns None if there is no such position.
EXAMPLES:
sage: Word('12113').conjugate_position(Word('31211')) 1 sage: Word('12131').conjugate_position(Word('12113')) is None True sage: Word().conjugate_position(Word('123')) is None True
TESTS:
We check that trac ticket #11128 is fixed:
sage: w = Word([0,0,1,0,2,1]) sage: [w.conjugate(i).conjugate_position(w) for i in range(w.length())] [0, 1, 2, 3, 4, 5]
-
conjugates
()¶ Returns the list of unique conjugates of self.
EXAMPLES:
sage: Word(range(6)).conjugates() [word: 012345, word: 123450, word: 234501, word: 345012, word: 450123, word: 501234] sage: Word('cbbca').conjugates() [word: cbbca, word: bbcac, word: bcacb, word: cacbb, word: acbbc]
The result contains each conjugate only once:
sage: Word('abcabc').conjugates() [word: abcabc, word: bcabca, word: cabcab]
TESTS:
sage: Word().conjugates() [word: ] sage: Word('a').conjugates() [word: a]
-
conjugates_iterator
()¶ Returns an iterator over the conjugates of self.
EXAMPLES:
sage: it = Word(range(4)).conjugates_iterator() sage: for w in it: w word: 0123 word: 1230 word: 2301 word: 3012
-
count
(letter)¶ Counts the number of occurrences of letter in self.
EXAMPLES:
sage: Word('abbabaab').count('a') 4
-
critical_exponent
()¶ Returns the critical exponent of self.
The critical exponent of a word is the supremum of the order of all its (finite) factors. See [1].
Note
The implementation here uses the suffix tree to enumerate all the factors. It should be improved.
EXAMPLES:
sage: Word('aaba').critical_exponent() 2 sage: Word('aabaa').critical_exponent() 2 sage: Word('aabaaba').critical_exponent() 7/3 sage: Word('ab').critical_exponent() 1 sage: Word('aba').critical_exponent() 3/2 sage: words.ThueMorseWord()[:20].critical_exponent() 2
REFERENCES:
- [1] F. Dejean. Sur un théorème de Thue. J. Combinatorial Theory Ser. A 13:90–99, 1972.
-
crochemore_factorization
()¶ Returns the Crochemore factorization of self as an ordered list of factors.
The Crochemore factorization of a finite word
is the unique factorization:
of
with each
satisfying either: C1.
is a letter that does not appear in
; C2.
is the longest prefix of
that also has an occurrence beginning within
. See [1].
Note
This is not a very good implementation, and should be improved.
EXAMPLES:
sage: x = Word('abababb') sage: x.crochemore_factorization() (a, b, abab, b) sage: mul(x.crochemore_factorization()) == x True sage: y = Word('abaababacabba') sage: y.crochemore_factorization() (a, b, a, aba, ba, c, ab, ba) sage: mul(y.crochemore_factorization()) == y True sage: x = Word([0,1,0,1,0,1,1]) sage: x.crochemore_factorization() (0, 1, 0101, 1) sage: mul(x.crochemore_factorization()) == x True
REFERENCES:
- [1] M. Crochemore, Recherche linéaire d’un carré dans un mot, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 14 781–784.
-
defect
(f=None)¶ Returns the defect of self.
The defect of a finite word
is given by the difference between the maximum number of possible palindromic factors in a word of length
and the actual number of palindromic factors contained in
. It is well known that the maximum number of palindromic factors in
is
(see [DJP01]).
An optional involution on letters
f
can be given. In that case, the f-palindromic defect (or pseudopalindromic defect, or theta-palindromic defect) ofis returned. It is a generalization of defect to f-palindromes. More precisely, the defect is
, where
denotes the set of f-palindromic factors of
(including the empty word) and
is the number of pairs
such that
is a letter,
is not equal to
, and
or
occurs in
. In the case of usual palindromes (i.e., for
f
not given or equal to the identity),for all
. See [BHNR04] for usual palindromes and [Sta11] for f-palindromes.
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e.,equal to the identity.
OUTPUT:
- integer – If f is None, the palindromic defect of self;
- otherwise, the f-palindromic defect of self.
EXAMPLES:
sage: Word('ara').defect() 0 sage: Word('abcacba').defect() 1
It is known that Sturmian words (see [DJP01]) have zero defect:
sage: words.FibonacciWord()[:100].defect() 0 sage: sa = WordMorphism('a->ab,b->b') sage: sb = WordMorphism('a->a,b->ba') sage: w = (sa*sb*sb*sa*sa*sa*sb).fixed_point('a') sage: w[:30].defect() 0 sage: w[110:140].defect() 0
It is even conjectured that the defect of an aperiodic word which is a fixed point of a primitive morphism is either
or infinite (see [BBGL08]):
sage: w = words.ThueMorseWord() sage: w[:50].defect() 12 sage: w[:100].defect() 16 sage: w[:300].defect() 52
For generalized defect with an involution different from the identity, there is always a letter which is not a palindrome! This is the reason for the modification of the definition:
sage: f = WordMorphism('a->b,b->a') sage: Word('a').defect(f) 0 sage: Word('ab').defect(f) 0 sage: Word('aa').defect(f) 1 sage: Word('abbabaabbaababba').defect(f) 3
sage: f = WordMorphism('a->b,b->a,c->c') sage: Word('cabc').defect(f) 0 sage: Word('abcaab').defect(f) 2
Other examples:
sage: Word('000000000000').defect() 0 sage: Word('011010011001').defect() 2 sage: Word('0101001010001').defect() 0 sage: Word().defect() 0 sage: Word('abbabaabbaababba').defect() 2
REFERENCES:
[BBGL08] A. Blondin Massé, S. Brlek, A. Garon, and S. Labbé, Combinatorial properties of f -palindromes in the Thue-Morse sequence. Pure Math. Appl., 19(2-3):39–52, 2008. [BHNR04] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306. [DJP01] (1, 2) X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255, (2001), no. 1–2, 539–553. [Sta11] Š. Starosta, On Theta-palindromic Richness, Theoret. Comp. Sci. 412 (2011) 1111–1121
-
deg_inv_lex_less
(other, weights=None)¶ Returns True if the word self is degree inverse lexicographically less than other.
EXAMPLES:
sage: Word([1,2,4]).deg_inv_lex_less(Word([1,3,2])) False sage: Word([3,2,1]).deg_inv_lex_less(Word([1,2,3])) True
-
deg_lex_less
(other, weights=None)¶ Returns True if self is degree lexicographically less than other, and False otherwise. The weight of each letter in the ordered alphabet is given by weights, which defaults to [1, 2, 3, ...].
EXAMPLES:
sage: Word([1,2,3]).deg_lex_less(Word([1,3,2])) True sage: Word([3,2,1]).deg_lex_less(Word([1,2,3])) False sage: W = Words(range(5)) sage: W([1,2,4]).deg_lex_less(W([1,3,2])) False sage: Word("abba").deg_lex_less(Word("abbb"), dict(a=1,b=2)) True sage: Word("abba").deg_lex_less(Word("baba"), dict(a=1,b=2)) True sage: Word("abba").deg_lex_less(Word("aaba"), dict(a=1,b=2)) False sage: Word("abba").deg_lex_less(Word("aaba"), dict(a=1,b=0)) True
-
deg_rev_lex_less
(other, weights=None)¶ Returns True if self is degree reverse lexicographically less than other.
EXAMPLES:
sage: Word([3,2,1]).deg_rev_lex_less(Word([1,2,3])) False sage: Word([1,2,4]).deg_rev_lex_less(Word([1,3,2])) False sage: Word([1,2,3]).deg_rev_lex_less(Word([1,2,4])) True
-
degree
(weights=None)¶ Returns the weighted degree of self, where the weighted degree of each letter in the ordered alphabet is given by weights, which defaults to [1, 2, 3, ...].
INPUTS:
weights
- a list or tuple, or a dictionary keyed by the letters occurring in self.
EXAMPLES:
sage: Word([1,2,3]).degree() 6 sage: Word([3,2,1]).degree() 6 sage: Words("ab")("abba").degree() 6 sage: Words("ab")("abba").degree([0,2]) 4 sage: Words("ab")("abba").degree([-1,-1]) -4 sage: Words("ab")("aabba").degree([1,1]) 5 sage: Words([1,2,4])([1,2,4]).degree() 6 sage: Word([1,2,4]).degree() 7 sage: Word("aabba").degree({'a':1,'b':2}) 7 sage: Word([0,1,0]).degree({0:17,1:0}) 34
-
delta
()¶ Returns the image of self under the delta morphism. This is the word composed of the length of consecutive runs of the same letter in a given word.
EXAMPLES:
sage: W = Words('0123456789') sage: W('22112122').delta() word: 22112 sage: W('555008').delta() word: 321 sage: W().delta() word: sage: Word('aabbabaa').delta() word: 22112
-
delta_derivate
(W=None)¶ Returns the derivative under delta for self.
EXAMPLES:
sage: W = Words('12') sage: W('12211').delta_derivate() word: 22 sage: W('1').delta_derivate(Words([1])) word: 1 sage: W('2112').delta_derivate() word: 2 sage: W('2211').delta_derivate() word: 22 sage: W('112').delta_derivate() word: 2 sage: W('11222').delta_derivate(Words([1, 2, 3])) word: 3
-
delta_derivate_left
(W=None)¶ Returns the derivative under delta for self.
EXAMPLES:
sage: W = Words('12') sage: W('12211').delta_derivate_left() word: 22 sage: W('1').delta_derivate_left(Words([1])) word: 1 sage: W('2112').delta_derivate_left() word: 21 sage: W('2211').delta_derivate_left() word: 22 sage: W('112').delta_derivate_left() word: 21 sage: W('11222').delta_derivate_left(Words([1, 2, 3])) word: 3
-
delta_derivate_right
(W=None)¶ Returns the right derivative under delta for self.
EXAMPLES:
sage: W = Words('12') sage: W('12211').delta_derivate_right() word: 122 sage: W('1').delta_derivate_right(Words([1])) word: 1 sage: W('2112').delta_derivate_right() word: 12 sage: W('2211').delta_derivate_right() word: 22 sage: W('112').delta_derivate_right() word: 2 sage: W('11222').delta_derivate_right(Words([1, 2, 3])) word: 23
-
delta_inv
(W=None, s=None)¶ Lifts self via the delta operator to obtain a word containing the letters in alphabet (default is [0, 1]). The letters used in the construction start with s (default is alphabet[0]) and cycle through alphabet.
INPUT:
alphabet
- an iterables
- an object in the iterable
EXAMPLES:
sage: W = Words([1, 2]) sage: W([2, 2, 1, 1]).delta_inv() word: 112212 sage: W([1, 1, 1, 1]).delta_inv(Words('123')) word: 1231 sage: W([2, 2, 1, 1, 2]).delta_inv(s=2) word: 22112122
-
evaluation
(alphabet=None)¶ Return the abelian vector of self counting the occurrences of each letter.
The vector is defined w.r.t the order of the alphabet of the parent. See also
evaluation_dict()
.INPUT:
self
– word having a parent on a finite alphabetalphabet
– DEPRECATED
OUTPUT:
listEXAMPLES:
sage: W = Words('ab') sage: W('aaabbbbb').abelian_vector() [3, 5] sage: W('a').abelian_vector() [1, 0] sage: W().abelian_vector() [0, 0]
The argument alphabet is deprecated:
sage: Word('aabaa').abelian_vector('abc') doctest:...: DeprecationWarning: The argument alphabet of methods abelian_vector and parikh_vector is deprecated and will be removed in a future version of Sage. In order to fix this, you must define your word on a parent with a finite alphabet. See http://trac.sagemath.org/17058 for details. [4, 1, 0]
You may fix the above deprecated use of alphabet argument this way:
sage: W = Words('abc') sage: W('aabaa').abelian_vector() [4, 1, 0]
TESTS:
sage: W = Words() sage: W('aabaa').abelian_vector() Traceback (most recent call last): ... TypeError: The alphabet of the parent is infinite; define the word with a parent on a finite alphabet or use evaluation_dict() instead
-
evaluation_dict
()¶ Returns a dictionary keyed by the letters occurring in self with values the number of occurrences of the letter.
EXAMPLES:
sage: Word([2,1,4,2,3,4,2]).evaluation_dict() {1: 1, 2: 3, 3: 1, 4: 2} sage: Word('badbcdb').evaluation_dict() {'a': 1, 'b': 3, 'c': 1, 'd': 2} sage: Word().evaluation_dict() {}
sage: f = Word('1213121').evaluation_dict() # keys appear in random order {'1': 4, '2': 2, '3': 1}
TESTS:
sage: f = Word('1213121').evaluation_dict() sage: f['1'] == 4 True sage: f['2'] == 2 True sage: f['3'] == 1 True
-
evaluation_partition
()¶ Returns the evaluation of the word w as a partition.
EXAMPLES:
sage: Word("acdabda").evaluation_partition() [3, 2, 1, 1] sage: Word([2,1,4,2,3,4,2]).evaluation_partition() [3, 2, 1, 1]
-
evaluation_sparse
()¶ Returns a list representing the evaluation of self. The entries of the list are two-element lists [a, n], where a is a letter occurring in self and n is the number of occurrences of a in self.
EXAMPLES:
sage: Word([4,4,2,5,2,1,4,1]).evaluation_sparse() [(1, 2), (2, 2), (4, 3), (5, 1)] sage: Word("abcaccab").evaluation_sparse() [('a', 3), ('c', 3), ('b', 2)]
-
exponent
()¶ Returns the exponent of self.
OUTPUT:
integer – the exponentEXAMPLES:
sage: Word('1231').exponent() 1 sage: Word('121212').exponent() 3 sage: Word().exponent() 0
-
factor_iterator
(n=None)¶ Generates distinct factors of
self
.INPUT:
n
- an integer, orNone
.
OUTPUT:
Ifn
is an integer, returns an iterator over all distinct factors of lengthn
. Ifn
isNone
, returns an iterator generating all distinct factors.EXAMPLES:
sage: w = Word('1213121') sage: sorted( w.factor_iterator(0) ) [word: ] sage: sorted( w.factor_iterator(10) ) [] sage: sorted( w.factor_iterator(1) ) [word: 1, word: 2, word: 3] sage: sorted( w.factor_iterator(4) ) [word: 1213, word: 1312, word: 2131, word: 3121] sage: sorted( w.factor_iterator() ) [word: , word: 1, word: 12, word: 121, word: 1213, word: 12131, word: 121312, word: 1213121, word: 13, word: 131, word: 1312, word: 13121, word: 2, word: 21, word: 213, word: 2131, word: 21312, word: 213121, word: 3, word: 31, word: 312, word: 3121]
sage: u = Word([1,2,1,2,3]) sage: sorted( u.factor_iterator(0) ) [word: ] sage: sorted( u.factor_iterator(10) ) [] sage: sorted( u.factor_iterator(1) ) [word: 1, word: 2, word: 3] sage: sorted( u.factor_iterator(5) ) [word: 12123] sage: sorted( u.factor_iterator() ) [word: , word: 1, word: 12, word: 121, word: 1212, word: 12123, word: 123, word: 2, word: 21, word: 212, word: 2123, word: 23, word: 3]
sage: xxx = Word("xxx") sage: sorted( xxx.factor_iterator(0) ) [word: ] sage: sorted( xxx.factor_iterator(4) ) [] sage: sorted( xxx.factor_iterator(2) ) [word: xx] sage: sorted( xxx.factor_iterator() ) [word: , word: x, word: xx, word: xxx]
sage: e = Word() sage: sorted( e.factor_iterator(0) ) [word: ] sage: sorted( e.factor_iterator(17) ) [] sage: sorted( e.factor_iterator() ) [word: ]
TESTS:
sage: type( Word('cacao').factor_iterator() ) <type 'generator'>
-
factor_occurrences_in
(other)¶ Returns an iterator over all occurrences (including overlapping ones) of self in other in their order of appearance.
EXAMPLES:
sage: u = Word('121') sage: w = Word('121213211213') sage: list(u.factor_occurrences_in(w)) [0, 2, 8]
-
factor_set
(n=None, algorithm='suffix tree')¶ Returns the set of factors (of length n) of self.
INPUT:
n
- an integer orNone
(default: None).algorithm
- string (default:'suffix tree'
), takes the following values:'suffix tree'
– construct and use the suffix tree of the word'naive'
– algorithm uses a sliding window
OUTPUT:
Ifn
is an integer, returns the set of all distinct factors of lengthn
. Ifn
isNone
, returns the set of all distinct factors.EXAMPLES:
sage: w = Word('121') sage: sorted(w.factor_set()) [word: , word: 1, word: 12, word: 121, word: 2, word: 21] sage: sorted(w.factor_set(algorithm='naive')) [word: , word: 1, word: 12, word: 121, word: 2, word: 21]
sage: w = Word('1213121') sage: for i in range(w.length()): sorted(w.factor_set(i)) [word: ] [word: 1, word: 2, word: 3] [word: 12, word: 13, word: 21, word: 31] [word: 121, word: 131, word: 213, word: 312] [word: 1213, word: 1312, word: 2131, word: 3121] [word: 12131, word: 13121, word: 21312] [word: 121312, word: 213121]
sage: w = Word([1,2,1,2,3]) sage: s = w.factor_set() sage: sorted(s) [word: , word: 1, word: 12, word: 121, word: 1212, word: 12123, word: 123, word: 2, word: 21, word: 212, word: 2123, word: 23, word: 3]
TESTS:
sage: w = Word("xx") sage: s = w.factor_set() sage: sorted(s) [word: , word: x, word: xx]
sage: Set(Word().factor_set()) {word: }
sage: w = Word(range(10), alphabet=range(10)) sage: S1 = w.factor_set(3, algorithm='suffix tree') sage: S2 = w.factor_set(3, algorithm='naive') sage: S1 == S2 True
-
find
(sub, start=0, end=None)¶ Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure.
INPUT:
sub
- string, list, tuple or word to search for.start
- non negative integer (default: 0) specifying the position from which to start the search.end
- non negative integer (default: None) specifying the position at which the search must stop. If None, then the search is performed up to the end of the string.
OUTPUT:
non negative integer or -1EXAMPLES:
sage: w = Word([0,1,0,0,1]) sage: w.find(Word([1,0])) 1
The
sub
argument can also be a tuple or a list:sage: w.find([1,0]) 1 sage: w.find((1,0)) 1
Examples using
start
andend
:sage: w.find(Word([0,1]), start=1) 3 sage: w.find(Word([0,1]), start=1, end=5) 3 sage: w.find(Word([0,1]), start=1, end=4) == -1 True sage: w.find(Word([1,1])) == -1 True sage: w.find("aa") -1
Instances of Word_str handle string inputs as well:
sage: w = Word('abac') sage: w.find('a') 0 sage: w.find('ba') 1
TESTS:
Check that trac ticket #12804 is fixed:
sage: w = Word(iter("ababab")) sage: w.find("ab") 0 sage: w.find("ab", start=1) 2 sage: w.find("aa") -1 sage: w.find("abc") -1 sage: w = Words('ab')(tuple('babaabaaab')) sage: w.find('abc') -1
-
first_pos_in
(other)¶ Returns the position of the first occurrence of self in other, or None if self is not a factor of other.
EXAMPLES:
sage: Word('12').first_pos_in(Word('131231')) 2 sage: Word('32').first_pos_in(Word('131231')) is None True
-
foata_bijection
()¶ Return word
self
under the Foata bijection.The Foata bijection
is a bijection on the set of words of given content (by a slight generalization of Section 2 in [FoSc78]). It can be defined by induction on the size of the word: Given a word
, start with
. At the
-th step, if
, we define
by placing
on the end of the word
and breaking the word up into blocks as follows. If
, place a vertical line to the right of each
for which
. Otherwise, if
, place a vertical line to the right of each
for which
. In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute
, the sequence of words is
,
,
,
,
,
,
.
So
.
See also
EXAMPLES:
sage: w = Word([2,2,2,1,1,1]) sage: w.foata_bijection() word: 112221 sage: w = Word([2,2,1,2,2,2,1,1,2,1]) sage: w.foata_bijection() word: 2122212211 sage: w = Word([4,1,5,4,2,2,3]) sage: w.foata_bijection() word: 1254423
TESTS:
sage: w = Word('121314') sage: w.foata_bijection() word: 231114 sage: w = Word('1133a1') sage: w.foata_bijection() word: 3113a1
-
good_suffix_table
()¶ Returns a table of the maximum skip you can do in order not to miss a possible occurrence of self in a word.
This is a part of the Boyer-Moore algorithm to find factors. See [1].
EXAMPLES:
sage: Word('121321').good_suffix_table() [5, 5, 5, 5, 3, 3, 1] sage: Word('12412').good_suffix_table() [3, 3, 3, 3, 3, 1]
REFERENCES:
- [1] R.S. Boyer, J.S. Moore, A fast string searching algorithm, Communications of the ACM 20 (1977) 762–772.
-
has_period
(p)¶ Returns True if self has the period
p
, False otherwise.Note
By convention, integers greater than the length of self are periods of self.
INPUT:
p
- an integer to check if it is a period of self.
EXAMPLES:
sage: w = Word('ababa') sage: w.has_period(2) True sage: w.has_period(3) False sage: w.has_period(4) True sage: w.has_period(-1) False sage: w.has_period(5) True sage: w.has_period(6) True
-
has_prefix
(other)¶ Test whether
self
hasother
as a prefix.INPUT:
other
- a word, or data describing a word
OUTPUT:
- boolean
EXAMPLES:
sage: w = Word("abbabaabababa") sage: u = Word("abbab") sage: w.has_prefix(u) True sage: u.has_prefix(w) False sage: u.has_prefix("abbab") True
sage: w = Word([0,1,1,0,1,0,0,1,0,1,0,1,0]) sage: u = Word([0,1,1,0,1]) sage: w.has_prefix(u) True sage: u.has_prefix(w) False sage: u.has_prefix([0,1,1,0,1]) True
-
has_suffix
(other)¶ Test whether
self
hasother
as a suffix.Note
Some word datatype classes, like
WordDatatype_str
, override this method.INPUT:
other
- a word, or data describing a word
OUTPUT:
- boolean
EXAMPLES:
sage: w = Word("abbabaabababa") sage: u = Word("ababa") sage: w.has_suffix(u) True sage: u.has_suffix(w) False sage: u.has_suffix("ababa") True
sage: w = Word([0,1,1,0,1,0,0,1,0,1,0,1,0]) sage: u = Word([0,1,0,1,0]) sage: w.has_suffix(u) True sage: u.has_suffix(w) False sage: u.has_suffix([0,1,0,1,0]) True
-
implicit_suffix_tree
()¶ Returns the implicit suffix tree of self.
The suffix tree of a word
is a compactification of the suffix trie for
. The compactification removes all nodes that have exactly one incoming edge and exactly one outgoing edge. It consists of two components: a tree and a word. Thus, instead of labelling the edges by factors of
, we can labelled them by indices of the occurrence of the factors in
.
See sage.combinat.words.suffix_trees.ImplicitSuffixTree? for more information.
EXAMPLES:
sage: w = Word("cacao") sage: w.implicit_suffix_tree() Implicit Suffix Tree of the word: cacao
sage: w = Word([0,1,0,1,1]) sage: w.implicit_suffix_tree() Implicit Suffix Tree of the word: 01011
-
inv_lex_less
(other)¶ Returns True if self is inverse lexicographically less than other.
EXAMPLES:
sage: Word([1,2,4]).inv_lex_less(Word([1,3,2])) False sage: Word([3,2,1]).inv_lex_less(Word([1,2,3])) True
-
inversions
()¶ Returns a list of the inversions of self. An inversion is a pair (i,j) of non-negative integers i < j such that self[i] > self[j].
EXAMPLES:
sage: Word([1,2,3,2,2,1]).inversions() [[1, 5], [2, 3], [2, 4], [2, 5], [3, 5], [4, 5]] sage: Words([3,2,1])([1,2,3,2,2,1]).inversions() [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2]] sage: Word('abbaba').inversions() [[1, 3], [1, 5], [2, 3], [2, 5], [4, 5]] sage: Words('ba')('abbaba').inversions() [[0, 1], [0, 2], [0, 4], [3, 4]]
-
is_balanced
(q=1)¶ Returns True if self is
-balanced, and False otherwise.
A finite or infinite word
is said to be
-balanced if for any two factors
,
of
of the same length, the difference between the number of
‘s in each of
and
is at most
for all letters
in the alphabet of
. A
-balanced word is simply said to be balanced. See for instance [1] and Chapter 2 of [2].
INPUT:
q
- integer (default 1), the balance level
OUTPUT:
boolean – the resultEXAMPLES:
sage: Word('1213121').is_balanced() True sage: Word('1122').is_balanced() False sage: Word('121333121').is_balanced() False sage: Word('121333121').is_balanced(2) False sage: Word('121333121').is_balanced(3) True sage: Word('121122121').is_balanced() False sage: Word('121122121').is_balanced(2) True
TESTS:
sage: Word('121122121').is_balanced(-1) Traceback (most recent call last): ... TypeError: the balance level must be a positive integer sage: Word('121122121').is_balanced(0) Traceback (most recent call last): ... TypeError: the balance level must be a positive integer sage: Word('121122121').is_balanced('a') Traceback (most recent call last): ... TypeError: the balance level must be a positive integer
REFERENCES:
- [1] J. Cassaigne, S. Ferenczi, L.Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble) 50 (2000) 1265–1276.
- [2] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
-
is_cadence
(seq)¶ Returns True if seq is a cadence of self, and False otherwise.
A cadence is an increasing sequence of indexes that all map to the same letter.
EXAMPLES:
sage: Word('121132123').is_cadence([0, 2, 6]) True sage: Word('121132123').is_cadence([0, 1, 2]) False sage: Word('121132123').is_cadence([]) True
-
is_conjugate_with
(other)¶ Returns True if self is a conjugate of other, and False otherwise.
INPUT:
other
- a finite word
OUPUT
bool
EXAMPLES:
sage: w = Word([0..20]) sage: z = Word([7..20] + [0..6]) sage: w word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 sage: z word: 7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6 sage: w.is_conjugate_with(z) True sage: z.is_conjugate_with(w) True sage: u = Word([4]*21) sage: u.is_conjugate_with(w) False sage: u.is_conjugate_with(z) False
Both words must be finite:
sage: w = Word(iter([2]*100),length='unknown') sage: z = Word([2]*100) sage: z.is_conjugate_with(w) #TODO: Not implemented for word of unknown length True sage: wf = Word(iter([2]*100),length='finite') sage: z.is_conjugate_with(wf) True sage: wf.is_conjugate_with(z) True
TESTS:
sage: Word('11213').is_conjugate_with(Word('31121')) True sage: Word().is_conjugate_with(Word('123')) False sage: Word('112131').is_conjugate_with(Word('11213')) False sage: Word('12131').is_conjugate_with(Word('11213')) True
We make sure that trac ticket #11128 is fixed:
sage: Word('abaa').is_conjugate_with(Word('aaba')) True sage: Word('aaba').is_conjugate_with(Word('abaa')) True
-
is_cube
()¶ Returns True if self is a cube, and False otherwise.
EXAMPLES:
sage: Word('012012012').is_cube() True sage: Word('01010101').is_cube() False sage: Word().is_cube() True sage: Word('012012').is_cube() False
-
is_cube_free
()¶ Returns True if self does not contain cubes, and False otherwise.
EXAMPLES:
sage: Word('12312').is_cube_free() True sage: Word('32221').is_cube_free() False sage: Word().is_cube_free() True
TESTS:
We make sure that trac ticket #8490 is fixed:
sage: Word('111').is_cube_free() False sage: Word('2111').is_cube_free() False sage: Word('32111').is_cube_free() False
-
is_empty
()¶ Returns True if the length of self is zero, and False otherwise.
EXAMPLES:
sage: Word([]).is_empty() True sage: Word('a').is_empty() False
-
is_factor
(other)¶ Returns True if self is a factor of other, and False otherwise.
EXAMPLES:
sage: u = Word('2113') sage: w = Word('123121332131233121132123') sage: u.is_factor(w) True sage: u = Word('321') sage: w = Word('1231241231312312312') sage: u.is_factor(w) False
The empty word is factor of another word:
sage: Word().is_factor(Word()) True sage: Word().is_factor(Word('a')) True sage: Word().is_factor(Word([1,2,3])) True sage: Word().is_factor(Word(lambda n:n, length=5)) True
-
is_finite
()¶ Returns True.
EXAMPLES:
sage: Word([]).is_finite() True sage: Word('a').is_finite() True
-
is_full
(f=None)¶ Returns True if self has defect 0, and False otherwise.
A word is full (or rich) if its defect is zero (see [1]). If
f
is given, then the f-palindromic defect is used (see [2]).INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
- boolean – If f is None, whether self is full;
- otherwise, whether self is full of
-palindromes.
EXAMPLES:
sage: words.ThueMorseWord()[:100].is_full() False sage: words.FibonacciWord()[:100].is_full() True sage: Word('000000000000000').is_full() True sage: Word('011010011001').is_full() False sage: Word('2194').is_full() True sage: Word().is_full() True
sage: f = WordMorphism('a->b,b->a') sage: Word().is_full(f) True sage: w = Word('ab') sage: w.is_full() True sage: w.is_full(f) True
sage: f = WordMorphism('a->b,b->a') sage: Word('abab').is_full(f) True sage: Word('abba').is_full(f) False
A simple example of an infinite word full of f-palindromes:
sage: p = WordMorphism({0:'abc',1:'ab'}) sage: f = WordMorphism('a->b,b->a,c->c') sage: p(words.FibonacciWord()[:50]).is_full(f) True sage: p(words.FibonacciWord()[:150]).is_full(f) True
REFERENCES:
- [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
- [2] E. Pelantová, Š. Starosta, Infinite words rich and almost rich in generalized palindromes, in: G. Mauri, A. Leporati (Eds.), Developments in Language Theory, volume 6795 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 406–416
-
is_lyndon
()¶ Returns True if self is a Lyndon word, and False otherwise.
A Lyndon word is a non-empty word that is lexicographically smaller than all of its proper suffixes for the given order on its alphabet. That is,
is a Lyndon word if
is non-empty and for each factorization
(with
,
both non-empty), we have
.
Equivalently,
is a Lyndon word iff
is a non-empty word that is lexicographically smaller than all of its proper conjugates for the given order on its alphabet.
See for instance [1].
EXAMPLES:
sage: Word('123132133').is_lyndon() True sage: Word().is_lyndon() True sage: Word('122112').is_lyndon() False
TESTS:
A sanity check:
LyndonWords
generates Lyndon words, so we filter all words of lengthon the alphabet [1,2,3] for Lyndon words, and compare with the
LyndonWords
generator:sage: for n in range(1,10): ... lw1 = [w for w in Words([1,2,3], n) if w.is_lyndon()] ... lw2 = LyndonWords(3,n) ... if set(lw1) != set(lw2): print False
Filter all words of length 8 on the alphabet [c,a,b] for Lyndon words, and compare with the
LyndonWords
generator after mapping [a,b,c] to [2,3,1]:sage: lw = [w for w in Words('cab', 8) if w.is_lyndon()] sage: phi = WordMorphism({'a':2,'b':3,'c':1}) sage: set(map(phi, lw)) == set(LyndonWords(3,8)) True
REFERENCES:
- [1] M. Lothaire, Combinatorics On Words, vol. 17 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Massachusetts, 1983.
-
is_overlap
()¶ Returns True if self is an overlap, and False otherwise.
EXAMPLES:
sage: Word('12121').is_overlap() True sage: Word('123').is_overlap() False sage: Word('1231').is_overlap() False sage: Word('123123').is_overlap() False sage: Word('1231231').is_overlap() True sage: Word().is_overlap() False
-
is_palindrome
(f=None)¶ Returns True if self is a palindrome (or a
-palindrome), and False otherwise.
Let
be an involution that extends to a morphism on
. We say that
is a `f`-palindrome if
[1]. Also called `f`-pseudo-palindrome [2].
INPUT:
f
- involution (default:None
) on the alphabet of self. It must be callable on letters as well as words (e.g.WordMorphism
). The default value corresponds to usual palindromes, i.e.,equal to the identity.
EXAMPLES:
sage: Word('esope reste ici et se repose').is_palindrome() False sage: Word('esoperesteicietserepose').is_palindrome() True sage: Word('I saw I was I').is_palindrome() True sage: Word('abbcbba').is_palindrome() True sage: Word('abcbdba').is_palindrome() False
Some
-palindromes:
sage: f = WordMorphism('a->b,b->a') sage: Word('aababb').is_palindrome(f) True
sage: f = WordMorphism('a->b,b->a,c->c') sage: Word('abacbacbab').is_palindrome(f) True
sage: f = WordMorphism({'a':'b','b':'a'}) sage: Word('aababb').is_palindrome(f) True
sage: f = WordMorphism({0:[1],1:[0]}) sage: w = words.ThueMorseWord()[:8]; w word: 01101001 sage: w.is_palindrome(f) True
The word must be in the domain of the involution:
sage: f = WordMorphism('a->a') sage: Word('aababb').is_palindrome(f) Traceback (most recent call last): ... KeyError: 'b'
TESTS:
If the given involution is not an involution:
sage: f = WordMorphism('a->b,b->b') sage: Word('abab').is_palindrome(f) Traceback (most recent call last): ... TypeError: self (=a->b, b->b) is not an endomorphism
sage: Y = Word sage: Y().is_palindrome() True sage: Y('a').is_palindrome() True sage: Y('ab').is_palindrome() False sage: Y('aba').is_palindrome() True sage: Y('aa').is_palindrome() True sage: E = WordMorphism('a->b,b->a') sage: Y().is_palindrome(E) True sage: Y('a').is_palindrome(E) False sage: Y('ab').is_palindrome(E) True sage: Y('aa').is_palindrome(E) False sage: Y('aba').is_palindrome(E) False sage: Y('abab').is_palindrome(E) True
REFERENCES:
- [1] S. Labbé, Propriétés combinatoires des
-palindromes, Mémoire de maîtrise en Mathématiques, Montréal, UQAM, 2008, 109 pages.
- [2] V. Anne, L.Q. Zamboni, I. Zorca, Palindromes and Pseudo- Palindromes in Episturmian and Pseudo-Palindromic Infinite Words, in : S. Brlek, C. Reutenauer (Eds.), Words 2005, Publications du LaCIM, Vol. 36 (2005) 91–100.
-
is_prefix
(other)¶ Returns True if self is a prefix of other, and False otherwise.
EXAMPLES:
sage: w = Word('0123456789') sage: y = Word('012345') sage: y.is_prefix(w) True sage: w.is_prefix(y) False sage: w.is_prefix(Word()) False sage: Word().is_prefix(w) True sage: Word().is_prefix(Word()) True
-
is_primitive
()¶ Returns True if self is primitive, and False otherwise.
A finite word
is primitive if it is not a positive integer power of a shorter word.
EXAMPLES:
sage: Word('1231').is_primitive() True sage: Word('111').is_primitive() False
-
is_proper_prefix
(other)¶ Returns True if self is a proper prefix of other, and False otherwise.
EXAMPLES:
sage: Word('12').is_proper_prefix(Word('123')) True sage: Word('12').is_proper_prefix(Word('12')) False sage: Word().is_proper_prefix(Word('123')) True sage: Word('123').is_proper_prefix(Word('12')) False sage: Word().is_proper_prefix(Word()) False
-
is_proper_suffix
(other)¶ Returns True if self is a proper suffix of other, and False otherwise.
EXAMPLES:
sage: Word('23').is_proper_suffix(Word('123')) True sage: Word('12').is_proper_suffix(Word('12')) False sage: Word().is_proper_suffix(Word('123')) True sage: Word('123').is_proper_suffix(Word('12')) False
-
is_quasiperiodic
()¶ Returns True if self is quasiperiodic, and False otherwise.
A finite or infinite word
is quasiperiodic if it can be constructed by concatenations and superpositions of one of its proper factors
, which is called a quasiperiod of
. See for instance [1], [2], and [3].
EXAMPLES:
sage: Word('abaababaabaababaaba').is_quasiperiodic() True sage: Word('abacaba').is_quasiperiodic() False sage: Word('a').is_quasiperiodic() False sage: Word().is_quasiperiodic() False sage: Word('abaaba').is_quasiperiodic() True
REFERENCES:
- [1] A. Apostolico, A. Ehrenfeucht, Efficient detection of quasiperiodicities in strings, Theoret. Comput. Sci. 119 (1993) 247–265.
- [2] S. Marcus, Quasiperiodic infinite words, Bull. Eur. Assoc. Theor. Comput. Sci. 82 (2004) 170-174.
- [3] A. Glen, F. Levé, G. Richomme, Quasiperiodic and Lyndon episturmian words, Preprint, 2008, arXiv:0805.0730.
-
is_rich
(f=None)¶ Returns True if self has defect 0, and False otherwise.
A word is full (or rich) if its defect is zero (see [1]). If
f
is given, then the f-palindromic defect is used (see [2]).INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
- boolean – If f is None, whether self is full;
- otherwise, whether self is full of
-palindromes.
EXAMPLES:
sage: words.ThueMorseWord()[:100].is_full() False sage: words.FibonacciWord()[:100].is_full() True sage: Word('000000000000000').is_full() True sage: Word('011010011001').is_full() False sage: Word('2194').is_full() True sage: Word().is_full() True
sage: f = WordMorphism('a->b,b->a') sage: Word().is_full(f) True sage: w = Word('ab') sage: w.is_full() True sage: w.is_full(f) True
sage: f = WordMorphism('a->b,b->a') sage: Word('abab').is_full(f) True sage: Word('abba').is_full(f) False
A simple example of an infinite word full of f-palindromes:
sage: p = WordMorphism({0:'abc',1:'ab'}) sage: f = WordMorphism('a->b,b->a,c->c') sage: p(words.FibonacciWord()[:50]).is_full(f) True sage: p(words.FibonacciWord()[:150]).is_full(f) True
REFERENCES:
- [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
- [2] E. Pelantová, Š. Starosta, Infinite words rich and almost rich in generalized palindromes, in: G. Mauri, A. Leporati (Eds.), Developments in Language Theory, volume 6795 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 406–416
-
is_smooth_prefix
()¶ Returns True if self is the prefix of a smooth word, and False otherwise.
Let
,
. An infinite word
in
is said to be smooth if and only if for all positive integers
,
is in
, where
is the word obtained from
by composing the length of consecutive runs of the same letter in
. See for instance [1] and [2].
INPUT:
self
- must be a word over the integers to get something other than False
OUTPUT:
boolean – whether self is a smooth prefix or notEXAMPLES:
sage: W = Words([1, 2]) sage: W([1, 1, 2, 2, 1, 2, 1, 1]).is_smooth_prefix() True sage: W([1, 2, 1, 2, 1, 2]).is_smooth_prefix() False
REFERENCES:
- [1] S. Brlek, A. Ladouceur, A note on differentiable palindromes, Theoret. Comput. Sci. 302 (2003) 167–178.
- [2] S. Brlek, S. Dulucq, A. Ladouceur, L. Vuillon, Combinatorial properties of smooth infinite words, Theoret. Comput. Sci. 352 (2006) 306–317.
-
is_square
()¶ Returns True if self is a square, and False otherwise.
EXAMPLES:
sage: Word([1,0,0,1]).is_square() False sage: Word('1212').is_square() True sage: Word('1213').is_square() False sage: Word('12123').is_square() False sage: Word().is_square() True
-
is_square_free
()¶ Returns True if self does not contain squares, and False otherwise.
EXAMPLES:
sage: Word('12312').is_square_free() True sage: Word('31212').is_square_free() False sage: Word().is_square_free() True
TESTS:
We make sure that trac ticket #8490 is fixed:
sage: Word('11').is_square_free() False sage: Word('211').is_square_free() False sage: Word('3211').is_square_free() False
-
is_sturmian_factor
()¶ Tells whether
self
is a factor of a Sturmian word.The finite word
self
must be defined on a two-letter alphabet.Equivalently, tells whether
self
is balanced. The advantage over the is_balanced method is that this one runs in linear time whereas is_balanced runs in quadratic time.OUTPUT:
- boolean – the result.
EXAMPLES:
sage: w = Word('0111011011011101101',alphabet='01') sage: w.is_sturmian_factor() True
sage: words.LowerMechanicalWord(random(),alphabet='01')[:100].is_sturmian_factor() True sage: words.CharacteristicSturmianWord(random())[:100].is_sturmian_factor() True
sage: w = Word('aabb',alphabet='ab') sage: w.is_sturmian_factor() False sage: s1 = WordMorphism('a->ab,b->b') sage: s2 = WordMorphism('a->ba,b->b') sage: s3 = WordMorphism('a->a,b->ba') sage: s4 = WordMorphism('a->a,b->ab') sage: W = Words('ab') sage: w = W('ab') sage: for i in xrange(8): w = choice([s1,s2,s3,s4])(w) sage: w word: abaaabaaabaabaaabaaabaabaaabaabaaabaaaba... sage: w.is_sturmian_factor() True
Famous words:
sage: words.FibonacciWord()[:100].is_sturmian_factor() True sage: words.ThueMorseWord()[:1000].is_sturmian_factor() False sage: words.KolakoskiWord()[:1000].is_sturmian_factor() False
REFERENCES:
[Arn2002] P. Arnoux, Sturmian sequences, in Substitutions in Dynamics, N. Pytheas Fogg (Ed.), Arithmetics, and Combinatorics (Lecture Notes in Mathematics, Vol. 1794), 2002. [Ser1985] C. Series. The geometry of Markoff numbers. The Mathematical Intelligencer, 7(3):20–29, 1985. [SU2009] J. Smillie and C. Ulcigrai. Symbolic coding for linear trajectories in the regular octagon, Arxiv 0905.0871, 2009. AUTHOR:
- Thierry Monteil
-
is_subword_of
(other)¶ Returns True is self is a subword of other, and False otherwise.
EXAMPLES:
sage: Word().is_subword_of(Word('123')) True sage: Word('123').is_subword_of(Word('3211333213233321')) True sage: Word('321').is_subword_of(Word('11122212112122133111222332')) False
See also
-
is_suffix
(other)¶ Returns True if w is a suffix of other, and False otherwise.
EXAMPLES:
sage: w = Word('0123456789') sage: y = Word('56789') sage: y.is_suffix(w) True sage: w.is_suffix(y) False sage: Word('579').is_suffix(w) False sage: Word().is_suffix(y) True sage: w.is_suffix(Word()) False sage: Word().is_suffix(Word()) True
-
is_symmetric
(f=None)¶ Returns True if self is symmetric (or
-symmetric), and False otherwise.
A word is symmetric (resp.
-symmetric) if it is the product of two palindromes (resp.
-palindromes). See [1] and [2].
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
EXAMPLES:
sage: Word('abbabab').is_symmetric() True sage: Word('ababa').is_symmetric() True sage: Word('aababaabba').is_symmetric() False sage: Word('aabbbaababba').is_symmetric() False sage: f = WordMorphism('a->b,b->a') sage: Word('aabbbaababba').is_symmetric(f) True
REFERENCES:
- [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic Complexity of Infinite Words, in J. Berstel, J. Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications, International Journal of Foundation of Computer Science, Vol. 15, No. 2 (2004) 293–306.
- [2] A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
-
is_tangent
()¶ Tells whether
self
is a tangent word.The finite word
self
must be defined on a two-letter alphabet.A binary word is said to be tangent if it can appear in infintely many cutting sequences of a smooth curve, where each cutting sequence is observed on a progressively smaller grid.
This class of words strictly contains the class of 1-balanced words, and is strictly contained in the class of 2-balanced words.
This method runs in linear time.
OUTPUT:
- boolean – the result.
EXAMPLES:
sage: w = Word('01110110110111011101',alphabet='01') sage: w.is_tangent() True
Some tangent words may not be balanced:
sage: Word('aabb',alphabet='ab').is_balanced() False sage: Word('aabb',alphabet='ab').is_tangent() True
Some 2-balanced words may not be tangent:
sage: Word('aaabb',alphabet='ab').is_tangent() False sage: Word('aaabb',alphabet='ab').is_balanced(2) True
Famous words:
sage: words.FibonacciWord()[:100].is_tangent() True sage: words.ThueMorseWord()[:1000].is_tangent() True sage: words.KolakoskiWord()[:1000].is_tangent() False
REFERENCES:
[Mon2010] T. Monteil, The asymptotic language of smooth curves, talk at LaCIM2010. AUTHOR:
- Thierry Monteil
-
iterated_left_palindromic_closure
(f=None)¶ Returns the iterated left (
-)palindromic closure of self.
INPUT:
f
– involution (default:None
) on the alphabet ofself
. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
word – the left iterated
-palindromic closure of
self
.EXAMPLES:
sage: Word('123').iterated_left_palindromic_closure() word: 3231323 sage: f = WordMorphism('a->b,b->a') sage: Word('ab').iterated_left_palindromic_closure(f=f) word: abbaab sage: Word('aab').iterated_left_palindromic_closure(f=f) word: abbaabbaab
TESTS:
If
f
is not a involution:sage: f = WordMorphism('a->b,b->b') sage: Word('aab').iterated_left_palindromic_closure(f=f) Traceback (most recent call last): ... TypeError: self (=a->b, b->b) is not an endomorphism
REFERENCES:
- A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
-
lacunas
(f=None)¶ Returns the list of all the lacunas of self.
A lacuna is a position in a word where the longest (
-)palindromic suffix is not unioccurrent (see [1]).
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e.,equal to the identity.
OUTPUT:
list – list of all the lacunas of self.EXAMPLES:
sage: w = Word([0,1,1,2,3,4,5,1,13,3]) sage: w.lacunas() [7, 9] sage: words.ThueMorseWord()[:100].lacunas() [8, 9, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 96, 97, 98, 99] sage: f = WordMorphism({0:[1],1:[0]}) sage: words.ThueMorseWord()[:50].lacunas(f) [0, 2, 4, 12, 16, 17, 18, 19, 48, 49]
REFERENCES:
- [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena, Arezzo-Italia), 53–67.
-
last_position_dict
()¶ Returns a dictionary that contains the last position of each letter in self.
EXAMPLES:
sage: Word('1231232').last_position_dict() {'1': 3, '2': 6, '3': 5}
-
left_special_factors
(n=None)¶ Returns the left special factors (of length n).
A factor
of a word
is left special if there are two distinct letters
and
such that
and
are factors of
.
INPUT:
n
- integer (optional, default:None
). IfNone
, it returns all left special factors.
OUTPUT:
A list of words.
EXAMPLES:
sage: alpha, beta, x = 0.54, 0.294, 0.1415 sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40] sage: for i in range(5): print i, sorted(w.left_special_factors(i)) 0 [word: ] 1 [word: 0] 2 [word: 00, word: 01] 3 [word: 000, word: 010] 4 [word: 0000, word: 0101]
-
left_special_factors_iterator
(n=None)¶ Returns an iterator over the left special factors (of length n).
A factor
of a word
is left special if there are two distinct letters
and
such that
and
are factors of
.
INPUT:
n
- integer (optional, default: None). If None, it returns an iterator over all left special factors.
EXAMPLES:
sage: alpha, beta, x = 0.54, 0.294, 0.1415 sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40] sage: sorted(w.left_special_factors_iterator(3)) [word: 000, word: 010] sage: sorted(w.left_special_factors_iterator(4)) [word: 0000, word: 0101] sage: sorted(w.left_special_factors_iterator(5)) [word: 00000, word: 01010]
-
length
()¶ Returns the length of self.
TESTS:
sage: from sage.combinat.words.word import Word_class sage: w = Word(iter('abba'*40), length="finite") sage: w._len is None True sage: w.length() 160 sage: w = Word(iter('abba'), length=4) sage: w._len 4 sage: w.length() 4 sage: def f(n): ... return range(2,12,2)[n] sage: w = Word(f, length=5) sage: w.length() 5
-
length_border
()¶ Returns the length of the border of self.
The border of a word is the longest word that is both a proper prefix and a proper suffix of self.
EXAMPLES:
sage: Word('121').length_border() 1 sage: Word('1').length_border() 0 sage: Word('1212').length_border() 2 sage: Word('111').length_border() 2 sage: Word().length_border() is None True
-
length_maximal_palindrome
(j, m=None, f=None)¶ Returns the length of the longest palindrome centered at position j.
INPUT:
j
– rational, position of the symmetry axis of the palindrome. Must return an integer when doubled. It is an integer when the center of the palindrome is a letter.m
– integer (default: None), minimal length of palindrome, if known. The parity ofm
can’t be the same as the parity of2j
.f
– involution (default: None), on the alphabet. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
The length of the longest-palindrome centered at position j.
EXAMPLES:
sage: Word('01001010').length_maximal_palindrome(3/2) 0 sage: Word('01101001').length_maximal_palindrome(3/2) 4 sage: Word('01010').length_maximal_palindrome(j=3, f='0->1,1->0') 0 sage: Word('01010').length_maximal_palindrome(j=2.5, f='0->1,1->0') 4 sage: Word('0222220').length_maximal_palindrome(3, f='0->1,1->0,2->2') 5
sage: w = Word('abcdcbaxyzzyx') sage: w.length_maximal_palindrome(3) 7 sage: w.length_maximal_palindrome(3, 3) 7 sage: w.length_maximal_palindrome(3.5) 0 sage: w.length_maximal_palindrome(9.5) 6 sage: w.length_maximal_palindrome(9.5, 2) 6
TESTS:
These are wrong inputs:
sage: w.length_maximal_palindrome(9.6) Traceback (most recent call last): ... ValueError: j must be positive, inferior to length of self sage: w.length_maximal_palindrome(3, 2) Traceback (most recent call last): ... ValueError: (2*j-m-1)/2(=3/2) must be an integer, i.e., 2*j(=6) and m(=2) can't have the same parity sage: w.length_maximal_palindrome(9.5, 3) Traceback (most recent call last): ... ValueError: (2*j-m-1)/2(=15/2) must be an integer, i.e., 2*j(=19) and m(=3) can't have the same parity
-
lengths_lps
(f=None)¶ Returns the list of the length of the longest palindromic suffix (lps) for each non-empty prefix of self.
It corresponds to the function
defined in [1].
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
- list – list of the length of the longest palindromic
- suffix (lps) for each non-empty prefix of self.
EXAMPLES:
sage: Word().lengths_lps() [] sage: Word('a').lengths_lps() [1] sage: Word('aaa').lengths_lps() [1, 2, 3] sage: Word('abbabaabbaab').lengths_lps() [1, 1, 2, 4, 3, 3, 2, 4, 2, 4, 6, 8]
sage: f = WordMorphism('a->b,b->a') sage: Word('abbabaabbaab').lengths_lps(f) [0, 2, 0, 2, 2, 4, 6, 8, 4, 6, 4, 6]
sage: f = WordMorphism({5:[8],8:[5]}) sage: Word([5,8,5,5,8,8,5,5,8,8,5,8,5]).lengths_lps(f) [0, 2, 2, 0, 2, 4, 6, 4, 6, 8, 10, 12, 4]
REFERENCES:
- [1] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi, Reconstructing words from a fixed palindromic length sequence, Proc. TCS 2008, 5th IFIP International Conference on Theoretical Computer Science (September 8-10 2008, Milano, Italia), accepted.
-
lengths_maximal_palindromes
(f=None)¶ Returns the length of maximal palindromes centered at each position.
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
list – The length of the maximal palindrome (or-palindrome) with a given symmetry axis (letter or space between two letters).
EXAMPLES:
sage: Word('01101001').lengths_maximal_palindromes() [0, 1, 0, 1, 4, 1, 0, 3, 0, 3, 0, 1, 4, 1, 0, 1, 0] sage: Word('00000').lengths_maximal_palindromes() [0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0] sage: Word('0').lengths_maximal_palindromes() [0, 1, 0] sage: Word('').lengths_maximal_palindromes() [0] sage: Word().lengths_maximal_palindromes() [0] sage: f = WordMorphism('a->b,b->a') sage: Word('abbabaab').lengths_maximal_palindromes(f) [0, 0, 2, 0, 0, 0, 2, 0, 8, 0, 2, 0, 0, 0, 2, 0, 0]
-
lengths_unioccurrent_lps
(f=None)¶ Returns the list of the lengths of the unioccurrent longest (
)-palindromic suffixes (lps) for each non-empty prefix of self. No unioccurrent lps are indicated by None.
It corresponds to the function
defined in [1] and [2].
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e.,equal to the identity.
OUTPUT:
- list – list of the length of the unioccurrent longest palindromic
- suffix (lps) for each non-empty prefix of self. No unioccurrent lps are indicated by None.
EXAMPLES:
sage: w = Word([0,1,1,2,3,4,5,1,13,3]) sage: w.lengths_unioccurrent_lps() [1, 1, 2, 1, 1, 1, 1, None, 1, None] sage: f = words.FibonacciWord()[:20] sage: f.lengths_unioccurrent_lps() == f.lengths_lps() True sage: t = words.ThueMorseWord() sage: t[:20].lengths_unioccurrent_lps() [1, 1, 2, 4, 3, 3, 2, 4, None, None, 6, 8, 10, 12, 14, 16, 6, 8, 10, 12] sage: f = WordMorphism({1:[0],0:[1]}) sage: t[:15].lengths_unioccurrent_lps(f) [None, 2, None, 2, None, 4, 6, 8, 4, 6, 4, 6, None, 4, 6]
REFERENCES:
- [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena, Arezzo-Italia), 53–67.
- [2] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi, Reconstructing words from a fixed palindromic length sequence, Proc. TCS 2008, 5th IFIP International Conference on Theoretical Computer Science (September 8-10 2008, Milano, Italia), accepted.
-
letters
()¶ Return the list of letters that appear in this word, listed in the order of first appearance.
EXAMPLES:
sage: Word([0,1,1,0,1,0,0,1]).letters() [0, 1] sage: Word("cacao").letters() ['c', 'a', 'o']
TESTS:
sage: Word().letters() []
-
longest_common_subword
(other)¶ Returns a longest subword of
self
andother
.A subword of a word is a subset of the word’s letters, read in the order in which they appear in the word.
For more information, see Wikipedia article Longest_common_subsequence_problem.
INPUT:
other
– a word
ALGORITHM:
For any indices
, we compute the longest common subword
lcs[i,j]
ofand
. This can be easily obtained as the longest of
lcs[i-1,j]
lcs[i,j-1]
lcs[i-1,j-1]+self[i]
ifself[i]==other[j]
EXAMPLES:
sage: v1 = Word("abc") sage: v2 = Word("ace") sage: v1.longest_common_subword(v2) word: ac sage: w1 = Word("1010101010101010101010101010101010101010") sage: w2 = Word("0011001100110011001100110011001100110011") sage: w1.longest_common_subword(w2) word: 00110011001100110011010101010
TESTS:
sage: Word().longest_common_subword(Word()) word:
See also
-
longest_common_suffix
(other)¶ Returns the longest common suffix of self and other.
EXAMPLES:
sage: w = Word('112345678') sage: u = Word('1115678') sage: w.longest_common_suffix(u) word: 5678 sage: u.longest_common_suffix(u) word: 1115678 sage: u.longest_common_suffix(w) word: 5678 sage: w.longest_common_suffix(w) word: 112345678 sage: y = Word('549332345') sage: w.longest_common_suffix(y) word:
TESTS:
With the empty word:
sage: w.longest_common_suffix(Word()) word: sage: Word().longest_common_suffix(w) word: sage: Word().longest_common_suffix(Word()) word:
With an infinite word:
sage: t=words.ThueMorseWord('ab') sage: w.longest_common_suffix(t) Traceback (most recent call last): ... TypeError: other must be a finite word
-
lps
(f=None, l=None)¶ Returns the longest palindromic (or
-palindromic) suffix of self.
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).l
- integer (default: None) the length of the longest palindrome suffix of self[:-1], if known.
OUTPUT:
- word – If f is None, the longest palindromic suffix of self;
- otherwise, the longest f-palindromic suffix of self.
EXAMPLES:
sage: Word('0111').lps() word: 111 sage: Word('011101').lps() word: 101 sage: Word('6667').lps() word: 7 sage: Word('abbabaab').lps() word: baab sage: Word().lps() word: sage: f = WordMorphism('a->b,b->a') sage: Word('abbabaab').lps(f=f) word: abbabaab sage: w = Word('33412321') sage: w.lps(l=3) word: 12321 sage: Y = Word sage: w = Y('01101001') sage: w.lps(l=2) word: 1001 sage: w.lps() word: 1001 sage: w.lps(l=None) word: 1001 sage: Y().lps(l=2) Traceback (most recent call last): ... IndexError: list index out of range sage: v = Word('abbabaab') sage: pal = v[:0] sage: for i in range(1, v.length()+1): ... pal = v[:i].lps(l=pal.length()) ... pal ... word: a word: b word: bb word: abba word: bab word: aba word: aa word: baab sage: f = WordMorphism('a->b,b->a') sage: v = Word('abbabaab') sage: pal = v[:0] sage: for i in range(1, v.length()+1): ... pal = v[:i].lps(f=f, l=pal.length()) ... pal ... word: word: ab word: word: ba word: ab word: baba word: bbabaa word: abbabaab
-
lps_lengths
(f=None)¶ Returns the length of the longest palindromic suffix of each prefix.
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
list – The length of the longest palindromic (or-palindromic) suffix of each prefix of self.
EXAMPLES:
sage: Word('01101001').lps_lengths() [0, 1, 1, 2, 4, 3, 3, 2, 4] sage: Word('00000').lps_lengths() [0, 1, 2, 3, 4, 5] sage: Word('0').lps_lengths() [0, 1] sage: Word('').lps_lengths() [0] sage: Word().lps_lengths() [0] sage: f = WordMorphism('a->b,b->a') sage: Word('abbabaab').lps_lengths(f) [0, 0, 2, 0, 2, 2, 4, 6, 8]
-
lyndon_factorization
()¶ Returns the Lyndon factorization of self.
The Lyndon factorization of a finite word
is the unique factorization of
as a non-increasing product of Lyndon words, i.e.,
where each
is a Lyndon word and
. See for instance [1].
OUTPUT:
list – the list of factors obtainedEXAMPLES:
sage: Word('010010010001000').lyndon_factorization() (01, 001, 001, 0001, 0, 0, 0) sage: Words('10')('010010010001000').lyndon_factorization() (0, 10010010001000) sage: Word('abbababbaababba').lyndon_factorization() (abb, ababb, aababb, a) sage: Words('ba')('abbababbaababba').lyndon_factorization() (a, bbababbaaba, bba) sage: Word([1,2,1,3,1,2,1]).lyndon_factorization() (1213, 12, 1)
TESTS:
sage: Words('01')('').lyndon_factorization() () sage: Word('01').lyndon_factorization() (01) sage: Words('10')('01').lyndon_factorization() (0, 1) sage: lynfac = Word('abbababbaababba').lyndon_factorization() sage: [x.is_lyndon() for x in lynfac] [True, True, True, True] sage: lynfac = Words('ba')('abbababbaababba').lyndon_factorization() sage: [x.is_lyndon() for x in lynfac] [True, True, True] sage: w = words.ThueMorseWord()[:1000] sage: w == prod(w.lyndon_factorization()) True
REFERENCES:
- [1] J.-P. Duval, Factorizing words over an ordered alphabet, J. Algorithms 4 (1983) 363–381.
- [2] G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42.
-
major_index
(final_descent=False)¶ Return the major index of
self
.The major index of a word
is the sum of the descents of
.
With the
final_descent
option, the last position of a non-empty word is also considered as a descent.See also
EXAMPLES:
sage: w = Word([2,1,3,3,2]) sage: w.major_index() 5 sage: w = Word([2,1,3,3,2]) sage: w.major_index(final_descent=True) 10
-
minimal_period
()¶ Returns the period of self.
Let
be an alphabet. An integer
is a period of a word
where
if
for
. The smallest period of
is called the period of
. See Chapter 1 of [1].
EXAMPLES:
sage: Word('aba').minimal_period() 2 sage: Word('abab').minimal_period() 2 sage: Word('ababa').minimal_period() 2 sage: Word('ababaa').minimal_period() 5 sage: Word('ababac').minimal_period() 6 sage: Word('aaaaaa').minimal_period() 1 sage: Word('a').minimal_period() 1 sage: Word().minimal_period() 1
REFERENCES:
- [1] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
-
nb_factor_occurrences_in
(other)¶ Returns the number of times self appears as a factor in other.
EXAMPLES:
sage: Word().nb_factor_occurrences_in(Word('123')) Traceback (most recent call last): ... NotImplementedError: The factor must be non empty sage: Word('123').nb_factor_occurrences_in(Word('112332312313112332121123')) 4 sage: Word('321').nb_factor_occurrences_in(Word('11233231231311233221123')) 0
-
nb_subword_occurrences_in
(other)¶ Returns the number of times self appears in other as a subword.
EXAMPLES:
sage: Word().nb_subword_occurrences_in(Word('123')) Traceback (most recent call last): ... NotImplementedError: undefined value sage: Word('123').nb_subword_occurrences_in(Word('1133432311132311112')) 11 sage: Word('4321').nb_subword_occurrences_in(Word('1132231112233212342231112')) 0 sage: Word('3').nb_subword_occurrences_in(Word('122332112321213')) 4
-
number_of_factors
(n=None, algorithm='suffix tree')¶ Counts the number of distinct factors of self.
INPUT:
n
- an integer, or None.algorithm
- string (default:'suffix tree'
), takes the following values:'suffix tree'
– construct and use the suffix tree of the word'naive'
– algorithm uses a sliding window
OUTPUT:
If n is an integer, returns the number of distinct factors of length n. If n is None, returns the total number of distinct factors.EXAMPLES:
sage: w = Word([1,2,1,2,3]) sage: w.number_of_factors() 13 sage: [w.number_of_factors(i) for i in range(6)] [1, 3, 3, 3, 2, 1]
sage: w = words.ThueMorseWord()[:100] sage: [w.number_of_factors(i) for i in range(10)] [1, 2, 4, 6, 10, 12, 16, 20, 22, 24]
sage: Word('1213121').number_of_factors() 22 sage: Word('1213121').number_of_factors(1) 3
sage: Word('a'*100).number_of_factors() 101 sage: Word('a'*100).number_of_factors(77) 1
sage: Word().number_of_factors() 1 sage: Word().number_of_factors(17) 0
sage: blueberry = Word("blueberry") sage: blueberry.number_of_factors() 43 sage: [blueberry.number_of_factors(i) for i in range(10)] [1, 6, 8, 7, 6, 5, 4, 3, 2, 1]
-
number_of_inversions
()¶ Return the number of inversions in
self
.An inversion of a word
is a pair of indices
with
and
.
See also
EXAMPLES:
sage: w = Word([2,1,3,3,2]) sage: w.number_of_inversions() 3
-
number_of_left_special_factors
(n)¶ Returns the number of left special factors of length n.
A factor
of a word
is left special if there are two distinct letters
and
such that
and
are factors of
.
INPUT:
n
- integer
OUTPUT:
Non negative integer
EXAMPLES:
sage: w = words.FibonacciWord()[:100] sage: [w.number_of_left_special_factors(i) for i in range(10)] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: w = words.ThueMorseWord()[:100] sage: [w.number_of_left_special_factors(i) for i in range(10)] [1, 2, 2, 4, 2, 4, 4, 2, 2, 4]
-
number_of_right_special_factors
(n)¶ Returns the number of right special factors of length n.
A factor
of a word
is right special if there are two distinct letters
and
such that
and
are factors of
.
INPUT:
n
- integer
OUTPUT:
Non negative integer
EXAMPLES:
sage: w = words.FibonacciWord()[:100] sage: [w.number_of_right_special_factors(i) for i in range(10)] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: w = words.ThueMorseWord()[:100] sage: [w.number_of_right_special_factors(i) for i in range(10)] [1, 2, 2, 4, 2, 4, 4, 2, 2, 4]
-
order
()¶ Returns the order of self.
Let
be the period of a word
. The positive rational number
is the order of
. See Chapter 8 of [1].
OUTPUT:
rational – the orderEXAMPLES:
sage: Word('abaaba').order() 2 sage: Word('ababaaba').order() 8/5 sage: Word('a').order() 1 sage: Word('aa').order() 2 sage: Word().order() 0
REFERENCES:
- [1] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
-
overlap_partition
(other, delay=0, p=None, involution=None)¶ Returns the partition of the alphabet induced by the overlap of self and other with the given delay.
The partition of the alphabet is given by the equivalence relation obtained from the symmetric, reflexive and transitive closure of the set of pairs of letters
where
,
are two words on the alphabet
and
is an integer.
The equivalence relation defined by
is inspired from [1].
INPUT:
other
- word on the same alphabet as selfdelay
- integerp
- disjoint sets data structure (optional, default: None), a partition of the alphabet into disjoint sets to start with. If None, each letter start in distinct equivalence classes.involution
- callable (optional, default: None), an involution on the alphabet. If involution is not None, the relationis considered.
OUTPUT:
- disjoint set data structure
EXAMPLES:
sage: W = Words(list('abc')+range(6)) sage: u = W('abc') sage: v = W(range(5)) sage: u.overlap_partition(v) {{0, 'a'}, {1, 'b'}, {2, 'c'}, {3}, {4}, {5}} sage: u.overlap_partition(v, 2) {{'a'}, {'b'}, {0, 'c'}, {1}, {2}, {3}, {4}, {5}} sage: u.overlap_partition(v, -1) {{0}, {1, 'a'}, {2, 'b'}, {3, 'c'}, {4}, {5}}
You can re-use the same disjoint set and do more than one overlap:
sage: p = u.overlap_partition(v, 2) sage: p {{'a'}, {'b'}, {0, 'c'}, {1}, {2}, {3}, {4}, {5}} sage: u.overlap_partition(v, 1, p) {{'a'}, {0, 1, 'b', 'c'}, {2}, {3}, {4}, {5}}
The function
overlap_partition
can be used to study equations on words. For example, if a wordoverlaps itself with delay
, then
is a period of
:
sage: W = Words(range(20)) sage: w = W(range(14)); w word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13 sage: d = 5 sage: p = w.overlap_partition(w, d) sage: m = WordMorphism(p.element_to_root_dict()) sage: w2 = m(w); w2 word: 56789567895678 sage: w2.minimal_period() == d True
If a word is equal to its reversal, then it is a palindrome:
sage: W = Words(range(20)) sage: w = W(range(17)); w word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 sage: p = w.overlap_partition(w.reversal(), 0) sage: m = WordMorphism(p.element_to_root_dict()) sage: w2 = m(w); w2 word: 01234567876543210 sage: w2.parent() Words over {0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 18, 19} sage: w2.is_palindrome() True
If the reversal of a word
is factor of its square
, then
is symmetric, i.e. the product of two palindromes:
sage: W = Words(range(10)) sage: w = W(range(10)); w word: 0123456789 sage: p = (w*w).overlap_partition(w.reversal(), 4) sage: m = WordMorphism(p.element_to_root_dict()) sage: w2 = m(w); w2 word: 0110456654 sage: w2.is_symmetric() True
If the image of the reversal of a word
under an involution
is factor of its square
, then
is
-symmetric:
sage: W = Words([-11,-9,..,11]) sage: w = W([1,3,..,11]) sage: w word: 1,3,5,7,9,11 sage: inv = lambda x:-x sage: f = WordMorphism(dict( (a, inv(a)) for a in W.alphabet())) sage: p = (w*w).overlap_partition(f(w).reversal(), 2, involution=f) sage: m = WordMorphism(p.element_to_root_dict()) sage: m(w) word: 1,-1,5,7,-7,-5 sage: m(w).is_symmetric(f) True
TESTS:
sage: W = Words('abcdef') sage: w = W('abc') sage: y = W('def') sage: w.overlap_partition(y, -3) {{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}} sage: w.overlap_partition(y, -2) {{'a', 'f'}, {'b'}, {'c'}, {'d'}, {'e'}} sage: w.overlap_partition(y, -1) {{'a', 'e'}, {'b', 'f'}, {'c'}, {'d'}} sage: w.overlap_partition(y, 0) {{'a', 'd'}, {'b', 'e'}, {'c', 'f'}} sage: w.overlap_partition(y, 1) {{'a'}, {'b', 'd'}, {'c', 'e'}, {'f'}} sage: w.overlap_partition(y, 2) {{'a'}, {'b'}, {'c', 'd'}, {'e'}, {'f'}} sage: w.overlap_partition(y, 3) {{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}} sage: w.overlap_partition(y, 4) {{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}
sage: W = Words(range(2)) sage: w = W([0,1,0,1,0,1]); w word: 010101 sage: w.overlap_partition(w, 0) {{0}, {1}} sage: w.overlap_partition(w, 1) {{0, 1}}
sage: empty = Word() sage: empty.overlap_partition(empty, 'yo') Traceback (most recent call last): ... TypeError: delay (=yo) must be an integer sage: empty.overlap_partition(empty,2,'yo') Traceback (most recent call last): ... TypeError: p(=yo) is not a DisjointSet
The involution input can be any callable:
sage: w = Words([-5,..,5])([-5..5]) sage: inv = lambda x:-x sage: w.overlap_partition(w, 2, involution=inv) {{-4, -2, 0, 2, 4}, {-5, -3, -1, 1, 3, 5}}
REFERENCES:
- [1] S. Labbé, Propriétés combinatoires des
-palindromes, Mémoire de maîtrise en Mathématiques, Montréal, UQAM, 2008, 109 pages.
-
palindrome_prefixes
()¶ Returns a list of all palindrome prefixes of self.
OUTPUT:
list – A list of all palindrome prefixes of self.EXAMPLES:
sage: w = Word('abaaba') sage: w.palindrome_prefixes() [word: , word: a, word: aba, word: abaaba] sage: w = Word('abbbbbbbbbb') sage: w.palindrome_prefixes() [word: , word: a]
-
palindromes
(f=None)¶ Returns the set of all palindromic (or
-palindromic) factors of self.
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
set - If f is None, the set of all palindromic factors of self; otherwise, the set of all f-palindromic factors of self.EXAMPLES:
sage: sorted(Word('01101001').palindromes()) [word: , word: 0, word: 00, word: 010, word: 0110, word: 1, word: 1001, word: 101, word: 11] sage: sorted(Word('00000').palindromes()) [word: , word: 0, word: 00, word: 000, word: 0000, word: 00000] sage: sorted(Word('0').palindromes()) [word: , word: 0] sage: sorted(Word('').palindromes()) [word: ] sage: sorted(Word().palindromes()) [word: ] sage: f = WordMorphism('a->b,b->a') sage: sorted(Word('abbabaab').palindromes(f)) [word: , word: ab, word: abbabaab, word: ba, word: baba, word: bbabaa]
-
palindromic_closure
(side='right', f=None)¶ Return the shortest palindrome having
self
as a prefix (or as a suffix ifside
is'left'
).See [1].
INPUT:
side
–'right'
or'left'
(default:'right'
) the direction of the closuref
– involution (default:None
) on the alphabet ofself
. It must be callable on letters as well as words (e.g. WordMorphism).
OUTPUT:
- word – If f is
None
, the right palindromic closure ofself
; - otherwise, the right
f
-palindromic closure ofself
. If side is'left'
, the left palindromic closure.
EXAMPLES:
sage: Word('1233').palindromic_closure() word: 123321 sage: Word('12332').palindromic_closure() word: 123321 sage: Word('0110343').palindromic_closure() word: 01103430110 sage: Word('0110343').palindromic_closure(side='left') word: 3430110343 sage: Word('01105678').palindromic_closure(side='left') word: 876501105678 sage: w = Word('abbaba') sage: w.palindromic_closure() word: abbababba
sage: f = WordMorphism('a->b,b->a') sage: w.palindromic_closure(f=f) word: abbabaab sage: w.palindromic_closure(f=f, side='left') word: babaabbaba
TESTS:
sage: f = WordMorphism('a->c,c->a') sage: w.palindromic_closure(f=f, side='left') Traceback (most recent call last): ... KeyError: 'b'
REFERENCES:
- [1] A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
-
palindromic_lacunas_study
(f=None)¶ Returns interesting statistics about longest (
-)palindromic suffixes and lacunas of self (see [1] and [2]).
Note that a word
has at most
different palindromic factors (see [3]). For
-palindromes (or pseudopalidromes or theta-palindromes), the maximum number of
-palindromic factors is
, where
is the number of pairs
such that
is a letter,
is not equal to
, and
or
occurs in
, see [4].
INPUT:
f
- involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism). The default value corresponds to usual palindromes, i.e.,equal to the identity.
OUTPUT:
list
- list of the length of the longest palindromic suffix (lps) for each non-empty prefix of self;list
- list of all the lacunas, i.e. positions where there is no unioccurrent lps;set
- set of palindromic factors of self.
EXAMPLES:
sage: a,b,c = Word('abbabaabbaab').palindromic_lacunas_study() sage: a [1, 1, 2, 4, 3, 3, 2, 4, 2, 4, 6, 8] sage: b [8, 9] sage: c # random order set([word: , word: b, word: bab, word: abba, word: bb, word: aa, word: baabbaab, word: baab, word: aba, word: aabbaa, word: a])
sage: f = WordMorphism('a->b,b->a') sage: a,b,c = Word('abbabaab').palindromic_lacunas_study(f=f) sage: a [0, 2, 0, 2, 2, 4, 6, 8] sage: b [0, 2, 4] sage: c # random order set([word: , word: ba, word: baba, word: ab, word: bbabaa, word: abbabaab]) sage: c == set([Word(), Word('ba'), Word('baba'), Word('ab'), Word('bbabaa'), Word('abbabaab')]) True
REFERENCES:
- [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena, Arezzo-Italia), 53–67.
- [2] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi, Reconstructing words from a fixed palindromic length sequence, Proc. TCS 2008, 5th IFIP International Conference on Theoretical Computer Science (September 8-10 2008, Milano, Italia), accepted.
- [3] X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001) 539–553.
- [4] Š. Starosta, On Theta-palindromic Richness, Theoret. Comp. Sci. 412 (2011) 1111–1121
-
parikh_vector
(*args, **kwds)¶ Deprecated: Use
abelian_vector()
instead. See trac ticket #17058 for details.
-
periods
(divide_length=False)¶ Returns a list containing the periods of self between
and
, where
is the length of self.
INPUT:
divide_length
- boolean (default: False). When set to True, then only periods that divide the length of self are considered.
OUTPUT:
List of positive integers
EXAMPLES:
sage: w = Word('ababab') sage: w.periods() [2, 4] sage: w.periods(divide_length=True) [2] sage: w = Word('ababa') sage: w.periods() [2, 4] sage: w.periods(divide_length=True) []
-
phi
()¶ Applies the phi function to self and returns the result. This is the word obtained by taking the first letter of the words obtained by iterating delta on self.
OUTPUT:
word – the result of the phi functionEXAMPLES:
sage: W = Words([1, 2]) sage: W([2,2,1,1,2,1,2,2,1,2,2,1,1,2]).phi() word: 222222 sage: W([2,1,2,2,1,2,2,1,2,1]).phi() word: 212113 sage: W().phi() word: sage: Word([2,1,2,2,1,2,2,1,2,1]).phi() word: 212113 sage: Word([2,3,1,1,2,1,2,3,1,2,2,3,1,2]).phi() word: 21215 sage: Word("aabbabaabaabba").phi() word: a22222 sage: w = Word([2,3,1,1,2,1,2,3,1,2,2,3,1,2])
REFERENCES:
- S. Brlek, A. Ladouceur, A note on differentiable palindromes, Theoret. Comput. Sci. 302 (2003) 167–178.
- S. Brlek, S. Dulucq, A. Ladouceur, L. Vuillon, Combinatorial properties of smooth infinite words, Theoret. Comput. Sci. 352 (2006) 306–317.
-
phi_inv
(W=None)¶ Apply the inverse of the phi function to self.
INPUT:
self
- a word over the integersW
- a parent object of words defined over integers
OUTPUT:
word – the inverse of the phi functionEXAMPLES:
sage: W = Words([1, 2]) sage: W([2, 2, 2, 2, 1, 2]).phi_inv() word: 22112122 sage: W([2, 2, 2]).phi_inv(Words([2, 3])) word: 2233
-
prefix_function_table
()¶ Returns a vector containing the length of the proper prefix-suffixes for all the non-empty prefixes of self.
EXAMPLES:
sage: Word('121321').prefix_function_table() [0, 0, 1, 0, 0, 1] sage: Word('1241245').prefix_function_table() [0, 0, 0, 1, 2, 3, 0] sage: Word().prefix_function_table() []
-
primitive
()¶ Returns the primitive of self.
EXAMPLES:
sage: Word('12312').primitive() word: 12312 sage: Word('121212').primitive() word: 12
-
primitive_length
()¶ Returns the length of the primitive of self.
EXAMPLES:
sage: Word('1231').primitive_length() 4 sage: Word('121212').primitive_length() 2
-
quasiperiods
()¶ Returns the quasiperiods of self as a list ordered from shortest to longest.
Let
be a finite or infinite word. A quasiperiod of
is a proper factor
of
such that the occurrences of
in
entirely cover
, i.e., every position of
falls within some occurrence of
in
. See for instance [1], [2], and [3].
EXAMPLES:
sage: Word('abaababaabaababaaba').quasiperiods() [word: aba, word: abaaba, word: abaababaaba] sage: Word('abaaba').quasiperiods() [word: aba] sage: Word('abacaba').quasiperiods() []
REFERENCES:
- [1] A. Apostolico, A. Ehrenfeucht, Efficient detection of quasiperiodicities in strings, Theoret. Comput. Sci. 119 (1993) 247–265.
- [2] S. Marcus, Quasiperiodic infinite words, Bull. Eur. Assoc. Theor. Comput. Sci. 82 (2004) 170-174.
- [3] A. Glen, F. Levé, G. Richomme, Quasiperiodic and Lyndon episturmian words, Preprint, 2008, arXiv:0805.0730.
-
rauzy_graph
(n)¶ Returns the Rauzy graph of the factors of length n of self.
The vertices are the factors of length
and there is an edge from
to
if
is a factor of length
for some letters
and
.
INPUT:
n
- integer
EXAMPLES:
sage: w = Word(range(10)); w word: 0123456789 sage: g = w.rauzy_graph(3); g Looped digraph on 8 vertices sage: WordOptions(identifier='') sage: g.vertices() [012, 123, 234, 345, 456, 567, 678, 789] sage: g.edges() [(012, 123, 3), (123, 234, 4), (234, 345, 5), (345, 456, 6), (456, 567, 7), (567, 678, 8), (678, 789, 9)] sage: WordOptions(identifier='word: ')
sage: f = words.FibonacciWord()[:100] sage: f.rauzy_graph(8) Looped digraph on 9 vertices
sage: w = Word('1111111') sage: g = w.rauzy_graph(3) sage: g.edges() [(word: 111, word: 111, word: 1)]
sage: w = Word('111') sage: for i in range(5) : w.rauzy_graph(i) Looped multi-digraph on 1 vertex Looped digraph on 1 vertex Looped digraph on 1 vertex Looped digraph on 1 vertex Looped digraph on 0 vertices
Multi-edges are allowed for the empty word:
sage: W = Words('abcde') sage: w = W('abc') sage: w.rauzy_graph(0) Looped multi-digraph on 1 vertex sage: _.edges() [(word: , word: , word: a), (word: , word: , word: b), (word: , word: , word: c)]
-
reduced_rauzy_graph
(n)¶ Returns the reduced Rauzy graph of order
of self.
INPUT:
n
- non negative integer. Every vertex of a reduced Rauzy graph of orderis a factor of length
of self.
OUTPUT:
Looped multi-digraph
DEFINITION:
For infinite periodic words (resp. for finite words of type
), the reduced Rauzy graph of order
(resp. for
smaller or equal to
) is the directed graph whose unique vertex is the prefix
of length
of self and which has an only edge which is a loop on
labelled by
where
is the unique return word to
.
In other cases, it is the directed graph defined as followed. Let
be the Rauzy graph of order
of self. The vertices are the vertices of
that are either special or not prolongable to the right or to the left. For each couple (
,
) of such vertices and each directed path in
from
to
that contains no other vertices that are special, there is an edge from
to
in the reduced Rauzy graph of order
whose label is the label of the path in
.
Note
In the case of infinite recurrent non periodic words, this definition correspond to the following one that can be found in [1] and [2] where a simple path is a path that begins with a special factor, ends with a special factor and contains no other vertices that are special:
The reduced Rauzy graph of factors of length
is obtained from
by replacing each simple path
with an edge
whose label is the concatenation of the labels of the edges of
.
EXAMPLES:
sage: w = Word(range(10)); w word: 0123456789 sage: g = w.reduced_rauzy_graph(3); g Looped multi-digraph on 2 vertices sage: g.vertices() [word: 012, word: 789] sage: g.edges() [(word: 012, word: 789, word: 3456789)]
For the Fibonacci word:
sage: f = words.FibonacciWord()[:100] sage: g = f.reduced_rauzy_graph(8);g Looped multi-digraph on 2 vertices sage: g.vertices() [word: 01001010, word: 01010010] sage: g.edges() [(word: 01001010, word: 01010010, word: 010), (word: 01010010, word: 01001010, word: 01010), (word: 01010010, word: 01001010, word: 10)]
For periodic words:
sage: from itertools import cycle sage: w = Word(cycle('abcd'))[:100] sage: g = w.reduced_rauzy_graph(3) sage: g.edges() [(word: abc, word: abc, word: dabc)]
sage: w = Word('111') sage: for i in range(5) : w.reduced_rauzy_graph(i) Looped digraph on 1 vertex Looped digraph on 1 vertex Looped digraph on 1 vertex Looped multi-digraph on 1 vertex Looped multi-digraph on 0 vertices
For ultimately periodic words:
sage: sigma = WordMorphism('a->abcd,b->cd,c->cd,d->cd') sage: w = sigma.fixed_point('a')[:100]; w word: abcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd... sage: g = w.reduced_rauzy_graph(5) sage: g.vertices() [word: abcdc, word: cdcdc] sage: g.edges() [(word: abcdc, word: cdcdc, word: dc), (word: cdcdc, word: cdcdc, word: dc)]
AUTHOR:
Julien Leroy (March 2010): initial version
REFERENCES:
- [1] M. Bucci et al. A. De Luca, A. Glen, L. Q. Zamboni, A connection between palindromic and factor complexity using return words,” Advances in Applied Mathematics 42 (2009) 60-74.
- [2] L’ubomira Balkova, Edita Pelantova, and Wolfgang Steiner. Sequences with constant number of return words. Monatsh. Math, 155 (2008) 251-263.
-
return_words
(fact)¶ Returns the set of return words of fact in self.
This is the set of all factors starting by the given factor and ending just before the next occurrence of this factor. See [1] and [2].
INPUT:
fact
- a non empty finite word
OUTPUT:
Python set of finite words
EXAMPLES:
sage: Word('21331233213231').return_words(Word('2')) {word: 213, word: 21331, word: 233} sage: Word().return_words(Word('213')) set() sage: Word('121212').return_words(Word('1212')) {word: 12}
sage: TM = words.ThueMorseWord()[:1000] sage: sorted(TM.return_words(Word([0]))) [word: 0, word: 01, word: 011]
REFERENCES:
- [1] F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998) 89-101.
- [2] C. Holton, L.Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Syst. 32 (1999) 133-157.
-
return_words_derivate
(fact)¶ Returns the word generated by mapping a letter to each occurrence of the return words for the given factor dropping any dangling prefix and suffix. See for instance [1].
EXAMPLES:
sage: Word('12131221312313122').return_words_derivate(Word('1')) word: 123242
REFERENCES:
- [1] F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998) 89–101.
-
rev_lex_less
(other)¶ Returns True if the word self is reverse lexicographically less than other.
EXAMPLES:
sage: Word([1,2,4]).rev_lex_less(Word([1,3,2])) True sage: Word([3,2,1]).rev_lex_less(Word([1,2,3])) False
-
reversal
()¶ Returns the reversal of self.
EXAMPLES:
sage: Word('124563').reversal() word: 365421
-
rfind
(sub, start=0, end=None)¶ Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure.
INPUT:
sub
- string, list, tuple or word to search for.start
- non negative integer (default: 0) specifying the position at which the search must stop.end
- non negative integer (default: None) specifying the position from which to start the search. If None, then the search is performed up to the end of the string.
OUTPUT:
non negative integer or -1EXAMPLES:
sage: w = Word([0,1,0,0,1]) sage: w.rfind(Word([0,1])) 3
The
sub
parameter can also be a list or a tuple:sage: w.rfind([0,1]) 3 sage: w.rfind((0,1)) 3
Examples using the argument
start
andend
:sage: w.rfind(Word([0,1]), end=4) 0 sage: w.rfind(Word([0,1]), end=5) 3 sage: w.rfind(Word([0,0]), start=2, end=5) 2 sage: w.rfind(Word([0,0]), start=3, end=5) -1
Instances of Word_str handle string inputs as well:
sage: w = Word('abac') sage: w.rfind('a') 2 sage: w.rfind(Word('a')) 2 sage: w.rfind([0,1]) -1
TESTS:
Check that trac ticket #12804 is fixed:
sage: w = Word(iter("abab")) sage: w.rfind("ab") 2 sage: w.rfind("ab", end=3) 0 sage: w.rfind("aa") -1 sage: w.rfind([0,0,0]) -1
-
right_special_factors
(n=None)¶ Returns the right special factors (of length n).
A factor
of a word
is right special if there are two distinct letters
and
such that
and
are factors of
.
INPUT:
n
- integer (optional, default: None). If None, it returns all right special factors.
OUTPUT:
A list of words.
EXAMPLES:
sage: w = words.ThueMorseWord()[:30] sage: for i in range(5): print i, sorted(w.right_special_factors(i)) 0 [word: ] 1 [word: 0, word: 1] 2 [word: 01, word: 10] 3 [word: 001, word: 010, word: 101, word: 110] 4 [word: 0110, word: 1001]
-
right_special_factors_iterator
(n=None)¶ Returns an iterator over the right special factors (of length n).
A factor
of a word
is right special if there are two distinct letters
and
such that
and
are factors of
.
INPUT:
n
- integer (optional, default: None). If None, it returns an iterator over all right special factors.
EXAMPLES:
sage: alpha, beta, x = 0.61, 0.54, 0.3 sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40] sage: sorted(w.right_special_factors_iterator(3)) [word: 010, word: 101] sage: sorted(w.right_special_factors_iterator(4)) [word: 0101, word: 1010] sage: sorted(w.right_special_factors_iterator(5)) [word: 00101, word: 11010]
-
robinson_schensted
()¶ Return the semistandard tableau and standard tableau pair obtained by running the Robinson-Schensted algorithm on
self
.This can also be done by running
RSK()
onself
.EXAMPLES:
sage: Word([1,1,3,1,2,3,1]).robinson_schensted() [[[1, 1, 1, 1, 3], [2], [3]], [[1, 2, 3, 5, 6], [4], [7]]]
-
schuetzenberger_involution
(n=None)¶ - Returns the Schuetzenberger involution of the word self, which is obtained by reverting the word and then complementing all letters within the underlying ordered alphabet. If
is specified, the underlying alphabet is assumed to be
. If no alphabet is specified,
is the maximal letter appearing in self.
INPUT:
self
– a wordn
– an integer specifying the maximal letter in the alphabet (optional)
OUTPUT:
- a word, the Schuetzenberger involution of self
EXAMPLES:
sage: w = Word([9,7,4,1,6,2,3]) sage: v = w.schuetzenberger_involution(); v word: 7849631 sage: v.parent() Words sage: w = Word([1,2,3],alphabet=[1,2,3,4,5]) sage: v = w.schuetzenberger_involution();v word: 345 sage: v.parent() Words over {1, 2, 3, 4, 5} sage: w = Word([1,2,3]) sage: v = w.schuetzenberger_involution(n=5);v word: 345 sage: v.parent() Words sage: w = Word([11,32,69,2,53,1,2,3,18,41]) sage: w.schuetzenberger_involution() word: 29,52,67,68,69,17,68,1,38,59 sage: w = Word([],alphabet=[1,2,3,4,5]) sage: w.schuetzenberger_involution() word: sage: w = Word([]) sage: w.schuetzenberger_involution() word:
-
shifted_shuffle
(other, shift=None)¶ Returns the combinatorial class representing the shifted shuffle product between words self and other. This is the same as the shuffle product of self with the word obtained from other by incrementing its values (i.e. its letters) by the given shift.
INPUT:
other
- finite word over the integersshift
- integer or None (default: None) added to each letter of other. When shift is None, it is replaced by self.length()
OUTPUT:
Combinatorial class of shifted shuffle products of self and other.EXAMPLES:
sage: w = Word([0,1,1]) sage: sp = w.shifted_shuffle(w); sp Shuffle product of word: 011 and word: 344 sage: sp = w.shifted_shuffle(w, 2); sp Shuffle product of word: 011 and word: 233 sage: sp.cardinality() 20 sage: WordOptions(identifier='') sage: sp.list() [011233, 012133, 012313, 012331, 021133, 021313, 021331, 023113, 023131, 023311, 201133, 201313, 201331, 203113, 203131, 203311, 230113, 230131, 230311, 233011] sage: WordOptions(identifier='word: ') sage: y = Word('aba') sage: y.shifted_shuffle(w,2) Traceback (most recent call last): ... ValueError: for shifted shuffle, words must only contain integers as letters
-
shuffle
(other, overlap=0)¶ Returns the combinatorial class representing the shuffle product between words self and other. This consists of all words of length self.length()+other.length() that have both self and other as subwords.
If overlap is non-zero, then the combinatorial class representing the shuffle product with overlaps is returned. The calculation of the shift in each overlap is done relative to the order of the alphabet. For example, “a” shifted by “a” is “b” in the alphabet [a, b, c] and 0 shifted by 1 in [0, 1, 2, 3] is 2.
INPUT:
other
- finite wordoverlap
- (default: 0) integer or True
OUTPUT:
Combinatorial class of shuffle product of self and otherEXAMPLES:
sage: ab = Word("ab") sage: cd = Word("cd") sage: sp = ab.shuffle(cd); sp Shuffle product of word: ab and word: cd sage: sp.cardinality() 6 sage: sp.list() [word: abcd, word: acbd, word: acdb, word: cabd, word: cadb, word: cdab] sage: w = Word([0,1]) sage: u = Word([2,3]) sage: w.shuffle(w) Shuffle product of word: 01 and word: 01 sage: u.shuffle(u) Shuffle product of word: 23 and word: 23 sage: w.shuffle(u) Shuffle product of word: 01 and word: 23 sage: w.shuffle(u,2) Overlapping shuffle product of word: 01 and word: 23 with 2 overlaps
-
standard_factorization
()¶ Returns the standard factorization of
self
.The standard factorization of a word
of length greater than 1 is the unique factorization:
where
is the longest proper suffix of
that is a Lyndon word.
Note that if
is a Lyndon word of length greater than 1 with standard factorization
, then
and
are also Lyndon words and
.
See for instance [1], [2] and [3].
INPUT:
self
- finite word of length greater than 1
OUTPUT:
tuple – tuple of two factorsEXAMPLES:
sage: Words('01')('0010110011').standard_factorization() (word: 001011, word: 0011) sage: Words('123')('1223312').standard_factorization() (word: 12233, word: 12) sage: Word([3,2,1]).standard_factorization() (word: 32, word: 1)
sage: w = Word('0010110011',alphabet='01') sage: w.standard_factorization() (word: 001011, word: 0011) sage: w = Word('0010110011',alphabet='10') sage: w.standard_factorization() (word: 001011001, word: 1) sage: w = Word('1223312',alphabet='123') sage: w.standard_factorization() (word: 12233, word: 12)
TESTS:
sage: w = Word() sage: w.standard_factorization() Traceback (most recent call last): ... ValueError: Standard factorization not defined on words of length less than 2 sage: w = Word('a') sage: w.standard_factorization() Traceback (most recent call last): ... ValueError: Standard factorization not defined on words of length less than 2
REFERENCES:
- [1] K.-T. Chen, R.H. Fox, R.C. Lyndon, Free differential calculus, IV. The quotient groups of the lower central series, Ann. of Math. 68 (1958) 81–95.
- [2] J.-P. Duval, Factorizing words over an ordered alphabet, J. Algorithms 4 (1983) 363–381.
- [3] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, U.K., 2002.
-
standard_permutation
()¶ Returns the standard permutation of the word self on the ordered alphabet. It is defined as the permutation with exactly the same number of inversions as w. Equivalently, it is the permutation of minimal length whose inverse sorts self.
EXAMPLES:
sage: w = Word([1,2,3,2,2,1]); w word: 123221 sage: p = w.standard_permutation(); p [1, 3, 6, 4, 5, 2] sage: v = Word(p.inverse().action(w)); v word: 112223 sage: filter(lambda q: q.length() <= p.length() and \ ....: q.inverse().action(w) == list(v), \ ....: Permutations(w.length()) ) [[1, 3, 6, 4, 5, 2]]
sage: w = Words([1,2,3])([1,2,3,2,2,1,2,1]); w word: 12322121 sage: p = w.standard_permutation(); p [1, 4, 8, 5, 6, 2, 7, 3] sage: Word(p.inverse().action(w)) word: 11122223
sage: w = Words([3,2,1])([1,2,3,2,2,1,2,1]); w word: 12322121 sage: p = w.standard_permutation(); p [6, 2, 1, 3, 4, 7, 5, 8] sage: Word(p.inverse().action(w)) word: 32222111
sage: w = Words('ab')('abbaba'); w word: abbaba sage: p = w.standard_permutation(); p [1, 4, 5, 2, 6, 3] sage: Word(p.inverse().action(w)) word: aaabbb
sage: w = Words('ba')('abbaba'); w word: abbaba sage: p = w.standard_permutation(); p [4, 1, 2, 5, 3, 6] sage: Word(p.inverse().action(w)) word: bbbaaa
-
sturmian_desubstitute_as_possible
()¶ Sturmian desubstitutes the word
self
as much as possible.The finite word
self
must be defined on a two-letter alphabet or use at most two-letters.It can be Sturmian desubstituted if one letter appears isolated: the Sturmian desubstitution consists in removing one letter per run of the non-isolated letter. The accelerated Sturmian desubstitution consists in removing a run equal to the length of the shortest inner run from any run of the non-isolated letter (including possible leading and trailing runs even if they have shorter length). The (accelerated) Sturmian desubstitution is done as much as possible. A word is a factor of a Sturmian word if, and only if, the result is the empty word.
OUTPUT:
- A finite word defined on a two-letter alphabet.
EXAMPLES:
sage: u = Word('10111101101110111',alphabet='01') ; u word: 10111101101110111 sage: v = u.sturmian_desubstitute_as_possible() ; v word: 01100101 sage: v == v.sturmian_desubstitute_as_possible() True sage: Word('azaazaaazaaazaazaaaz', alphabet='az').sturmian_desubstitute_as_possible() word:
TESTS:
sage: w = Word('azazaza', alphabet='aze') sage: w.sturmian_desubstitute_as_possible() word: sage: Word('aze').sturmian_desubstitute_as_possible() Traceback (most recent call last): ... TypeError: your word must be defined on a binary alphabet or use at most two different letters sage: Word('azaaazaazaazaaazaaza', alphabet='az').sturmian_desubstitute_as_possible() word: sage: Word('azaaazaazaazaaazaaaza', alphabet='az').sturmian_desubstitute_as_possible() word: azzaa
Boundary effects:
sage: Word('', alphabet='az').sturmian_desubstitute_as_possible() word: sage: Word('azzzzz', alphabet='az').sturmian_desubstitute_as_possible() word: sage: Word('zzzzza', alphabet='az').sturmian_desubstitute_as_possible() word: sage: Word('aaaaazaaaaaaaaa', alphabet='az').sturmian_desubstitute_as_possible() word: sage: Word('aaaaaaaaaaaaaa', alphabet='az').sturmian_desubstitute_as_possible() word:
Boundary effects without alphabet:
sage: Word('').sturmian_desubstitute_as_possible() word: sage: Word('azzzzz').sturmian_desubstitute_as_possible() word: sage: Word('zzzzza').sturmian_desubstitute_as_possible() word: sage: Word('aaaaazaaaaaaaaa').sturmian_desubstitute_as_possible() word: sage: Word('aaaaaaaaaaaaaa').sturmian_desubstitute_as_possible() word:
Idempotence:
sage: r = words.RandomWord(randint(1,15)).sturmian_desubstitute_as_possible() ; r == r.sturmian_desubstitute_as_possible() True
AUTHOR:
- Thierry Monteil
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suffix_tree
()¶ Alias for implicit_suffix_tree().
EXAMPLES:
sage: Word('abbabaab').suffix_tree() Implicit Suffix Tree of the word: abbabaab
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suffix_trie
()¶ Returns the suffix trie of self.
The suffix trie of a finite word
is a data structure representing the factors of
. It is a tree whose edges are labelled with letters of
, and whose leafs correspond to suffixes of
.
See sage.combinat.words.suffix_trees.SuffixTrie? for more information.
EXAMPLES:
sage: w = Word("cacao") sage: w.suffix_trie() Suffix Trie of the word: cacao
sage: w = Word([0,1,0,1,1]) sage: w.suffix_trie() Suffix Trie of the word: 01011
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swap
(i, j=None)¶ Returns the word w with entries at positions i and j swapped. By default, j = i+1.
EXAMPLES:
sage: Word([1,2,3]).swap(0,2) word: 321 sage: Word([1,2,3]).swap(1) word: 132 sage: Word("abba").swap(1,-1) word: aabb
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swap_decrease
(i)¶ Returns the word with positions i and i+1 exchanged if self[i] < self[i+1]. Otherwise, it returns self.
EXAMPLES:
sage: w = Word([1,3,2]) sage: w.swap_decrease(0) word: 312 sage: w.swap_decrease(1) word: 132 sage: w.swap_decrease(1) is w True sage: Words("ab")("abba").swap_decrease(0) word: baba sage: Words("ba")("abba").swap_decrease(0) word: abba
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swap_increase
(i)¶ Returns the word with positions i and i+1 exchanged if self[i] > self[i+1]. Otherwise, it returns self.
EXAMPLES:
sage: w = Word([1,3,2]) sage: w.swap_increase(1) word: 123 sage: w.swap_increase(0) word: 132 sage: w.swap_increase(0) is w True sage: Words("ab")("abba").swap_increase(0) word: abba sage: Words("ba")("abba").swap_increase(0) word: baba
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to_integer_list
()¶ Returns a list of integers from [0,1,...,self.length()-1] in the same relative order as the letters in self in the parent.
EXAMPLES:
sage: from itertools import count sage: w = Word('abbabaab') sage: w.to_integer_list() [0, 1, 1, 0, 1, 0, 0, 1] sage: w = Word(iter("cacao"), length="finite") sage: w.to_integer_list() [1, 0, 1, 0, 2] sage: w = Words([3,2,1])([2,3,3,1]) sage: w.to_integer_list() [1, 0, 0, 2]
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to_integer_word
()¶ Returns a word defined over the integers [0,1,...,self.length()-1] whose letters are in the same relative order in the parent.
EXAMPLES:
sage: from itertools import count sage: w = Word('abbabaab') sage: w.to_integer_word() word: 01101001 sage: w = Word(iter("cacao"), length="finite") sage: w.to_integer_word() word: 10102 sage: w = Words([3,2,1])([2,3,3,1]) sage: w.to_integer_word() word: 1002
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to_monoid_element
()¶ Return
self
as an element the free monoid with the same alphabet asself
.EXAMPLES:
sage: w = Word('aabb') sage: w.to_monoid_element() a^2*b^2 sage: W = Words('abc') sage: w = W(w) sage: w.to_monoid_element() a^2*b^2
TESTS:
Check that
w == w.to_monoid_element().to_word()
:sage: all(w.to_monoid_element().to_word() == w for i in range(6) for w in Words('abc', i)) True
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topological_entropy
(n)¶ Return the topological entropy for the factors of length n.
The topological entropy of a sequence
is defined as the exponential growth rate of the complexity of
as the length increases:
where
denotes the cardinality of the alphabet and
is the complexity function, i.e. the number of factors of length
in the sequence
[1].
INPUT:
self
- a word defined over a finite alphabetn
- positive integer
OUTPUT:
real number (a symbolic expression)
EXAMPLES:
sage: W = Words([0, 1]) sage: w = W([0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1]) sage: t = w.topological_entropy(3); t 1/3*log(7)/log(2) sage: n(t) 0.935784974019201
sage: w = words.ThueMorseWord()[:100] sage: topo = w.topological_entropy sage: for i in range(0, 41, 5): print i, n(topo(i), digits=5) 0 1.0000 5 0.71699 10 0.48074 15 0.36396 20 0.28774 25 0.23628 30 0.20075 35 0.17270 40 0.14827
If no alphabet is specified, an error is raised:
sage: w = Word(range(20)) sage: w.topological_entropy(3) Traceback (most recent call last): ... TypeError: The word must be defined over a finite alphabet
The following is ok:
sage: W = Words(range(20)) sage: w = W(range(20)) sage: w.topological_entropy(3) 1/3*log(18)/log(20)
REFERENCES:
[1] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics, and Combinatorics, Lecture Notes in Mathematics 1794, Springer Verlag. V. Berthe, S. Ferenczi, C. Mauduit and A. Siegel, Eds. (2002).
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