Common combinatorial tools¶
REFERENCES:
[NCSF] | (1, 2, 3, 4) Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon, Noncommutative Symmetric Functions, Adv. Math. 112 (1995), no. 2, 218-348. |
[QSCHUR] | Haglund, Luoto, Mason, van Willigenburg, Quasisymmetric Schur functions, J. Comb. Theory Ser. A 118 (2011), 463-490. http://www.sciencedirect.com/science/article/pii/S0097316509001745 , Arxiv 0810.2489v2. |
[Tev2007] | Lenny Tevlin, Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product, Arxiv 0712.2201v1. |
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sage.combinat.ncsf_qsym.combinatorics.
coeff_dab
(I, J)¶ Return the number of standard composition tableaux of shape
with descent composition
.
INPUT:
I, J
– compositions
OUTPUT:
- An integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_dab sage: coeff_dab(Composition([2,1]),Composition([2,1])) 1 sage: coeff_dab(Composition([1,1,2]),Composition([1,2,1])) 0
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sage.combinat.ncsf_qsym.combinatorics.
coeff_ell
(J, I)¶ Returns the coefficient
as defined in [NCSF].
INPUT:
J
– a compositionI
– a composition refiningJ
OUTPUT:
- integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_ell sage: coeff_ell(Composition([1,1,1]), Composition([2,1])) 2 sage: coeff_ell(Composition([2,1]), Composition([3])) 2
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sage.combinat.ncsf_qsym.combinatorics.
coeff_lp
(J, I)¶ Returns the coefficient
as defined in [NCSF].
INPUT:
J
– a compositionI
– a composition refiningJ
OUTPUT:
- integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_lp sage: coeff_lp(Composition([1,1,1]), Composition([2,1])) 1 sage: coeff_lp(Composition([2,1]), Composition([3])) 1
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sage.combinat.ncsf_qsym.combinatorics.
coeff_pi
(J, I)¶ Returns the coefficient
as defined in [NCSF].
INPUT:
J
– a compositionI
– a composition refiningJ
OUTPUT:
- integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_pi sage: coeff_pi(Composition([1,1,1]), Composition([2,1])) 2 sage: coeff_pi(Composition([2,1]), Composition([3])) 6
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sage.combinat.ncsf_qsym.combinatorics.
coeff_sp
(J, I)¶ Returns the coefficient
as defined in [NCSF].
INPUT:
J
– a compositionI
– a composition refiningJ
OUTPUT:
- integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_sp sage: coeff_sp(Composition([1,1,1]), Composition([2,1])) 2 sage: coeff_sp(Composition([2,1]), Composition([3])) 4
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sage.combinat.ncsf_qsym.combinatorics.
compositions_order
(n)¶ Return the compositions of
ordered as defined in [QSCHUR].
Let
return the composition
after sorting. For compositions
and
, we order
if
lexicographically, or
and
lexicographically.
INPUT:
n
– a positive integer
OUTPUT:
- A list of the compositions of
n
sorted into decreasing order by
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import compositions_order sage: compositions_order(3) [[3], [2, 1], [1, 2], [1, 1, 1]] sage: compositions_order(4) [[4], [3, 1], [1, 3], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
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sage.combinat.ncsf_qsym.combinatorics.
m_to_s_stat
(R, I, K)¶ Return the coefficient of the complete non-commutative symmetric function
in the expansion of the monomial non-commutative symmetric function
with respect to the complete basis over the ring
. This is the coefficient in formula (36) of Tevlin’s paper [Tev2007].
INPUT:
R
– A ring, supposed to be a-algebra
I
,K
– compositions
OUTPUT:
- The coefficient of
in the expansion of
in the complete basis of the non-commutative symmetric functions over
R
.
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import m_to_s_stat sage: m_to_s_stat(QQ, Composition([2,1]), Composition([1,1,1])) -1 sage: m_to_s_stat(QQ, Composition([3]), Composition([1,2])) -2 sage: m_to_s_stat(QQ, Composition([2,1,2]), Composition([2,1,2])) 8/3
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sage.combinat.ncsf_qsym.combinatorics.
number_of_SSRCT
(content_comp, shape_comp)¶ The number of semi-standard reverse composition tableaux.
The dual quasisymmetric-Schur functions satisfy a left Pieri rule where
is a sum over dual quasisymmetric-Schur functions indexed by compositions which contain the composition
. The definition of an SSRCT comes from this rule. The number of SSRCT of content
and shape
is equal to the number of SSRCT of content
and shape
where
appears in the expansion of
.
In sage the recording tableau for these objects are called
CompositionTableaux
.INPUT:
content_comp
,shape_comp
– compositions
OUTPUT:
- An integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import number_of_SSRCT sage: number_of_SSRCT(Composition([3,1]), Composition([1,3])) 0 sage: number_of_SSRCT(Composition([1,2,1]), Composition([1,3])) 1 sage: number_of_SSRCT(Composition([1,1,2,2]), Composition([3,3])) 2 sage: all(CompositionTableaux(be).cardinality() ....: == sum(number_of_SSRCT(al,be)*binomial(4,len(al)) ....: for al in Compositions(4)) ....: for be in Compositions(4)) True
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sage.combinat.ncsf_qsym.combinatorics.
number_of_fCT
(content_comp, shape_comp)¶ Return the number of Immaculate tableaux of shape
shape_comp
and contentcontent_comp
.See [BBSSZ2012], Definition 3.9, for the notion of an immaculate tableau.
INPUT:
content_comp
,shape_comp
– compositions
OUTPUT:
- An integer
EXAMPLES:
sage: from sage.combinat.ncsf_qsym.combinatorics import number_of_fCT sage: number_of_fCT(Composition([3,1]), Composition([1,3])) 0 sage: number_of_fCT(Composition([1,2,1]), Composition([1,3])) 1 sage: number_of_fCT(Composition([1,1,3,1]), Composition([2,1,3])) 2