These lattices depend on three parameters ,
and
, where
and
are
coprime positive integers and
is a nonnegative integer.
The elements are Dyck paths in the -rectangle. The order relation
depends on
.
To use the provided functionality, you should import Tamari lattices by typing:
sage: from sage.combinat.tamari_lattices import TamariLattice
or:
sage: from sage.combinat.tamari_lattices import GeneralizedTamariLattice
EXAMPLES:
sage: from sage.combinat.tamari_lattices import TamariLattice
sage: TamariLattice(3)
Finite lattice containing 5 elements
sage: from sage.combinat.tamari_lattices import GeneralizedTamariLattice
sage: GeneralizedTamariLattice(3,2)
Finite lattice containing 2 elements
sage: GeneralizedTamariLattice(4,3)
Finite lattice containing 5 elements
See also
For more detailed information see TamariLattice(), GeneralizedTamariLattice().
Returns the -Tamari lattice of parameter
.
INPUT:
OUTPUT:
The elements of the lattice are Dyck paths in the -rectangle.
The parameter (slope) is used only to define the covering relations.
When the slope
is
, two paths are comparable if and only if
one is always above the other.
The usual Tamari lattice of index is the special
case
and
.
Other special cases give the -Tamari lattices studied in [BMFPR].
EXAMPLES:
sage: from sage.combinat.tamari_lattices import GeneralizedTamariLattice
sage: GeneralizedTamariLattice(3,2)
Finite lattice containing 2 elements
sage: GeneralizedTamariLattice(4,3)
Finite lattice containing 5 elements
sage: GeneralizedTamariLattice(4,4)
Traceback (most recent call last):
...
ValueError: The numbers a and b must be coprime with a>=b.
sage: GeneralizedTamariLattice(7,5,2)
Traceback (most recent call last):
...
ValueError: The condition a>=b*m does not hold.
sage: P = GeneralizedTamariLattice(5,3);P
Finite lattice containing 7 elements
TESTS:
sage: P.coxeter_transformation()**18 == 1
True
REFERENCES:
[BMFPR] |
|
Returns the -th Tamari lattice.
INPUT:
OUTPUT:
The elements of the lattice are Dyck paths in the -rectangle.
See Tamari lattice for mathematical background.
EXAMPLES:
sage: from sage.combinat.tamari_lattices import TamariLattice
sage: TamariLattice(3)
Finite lattice containing 5 elements
Returns all Dyck paths from to
in the
-rectangle.
This means that at each step of the path, one has .
A path is represented by a sequence of and
, where
is an
horizontal step
and
is a vertical step
.
INPUT:
OUTPUT:
EXAMPLES:
sage: from sage.combinat.tamari_lattices import paths_in_triangle
sage: paths_in_triangle(2,2,2,2)
[(1, 0, 1, 0), (1, 1, 0, 0)]
sage: paths_in_triangle(2,3,4,4)
[(1, 0, 1, 0, 1), (1, 1, 0, 0, 1), (1, 0, 1, 1, 0), (1, 1, 0, 1, 0), (1, 1, 1, 0, 0)]
sage: paths_in_triangle(2,1,4,4)
Traceback (most recent call last):
...
ValueError: The endpoint is not valid.
sage: paths_in_triangle(3,2,5,3)
[(1, 0, 1, 0, 0), (1, 1, 0, 0, 0)]
Performs a covering move in the -Tamari lattice of parameter
.
The letter at position in
must be a
, followed by at
least one
.
INPUT:
OUTPUT:
EXAMPLES:
sage: from sage.combinat.tamari_lattices import swap
sage: swap((1,0,1,0),1)
(1, 1, 0, 0)
sage: swap((1,0,1,0),6)
Traceback (most recent call last):
...
ValueError: The index is greater than the length of the path.
sage: swap((1,1,0,0,1,1,0,0),3)
(1, 1, 0, 1, 1, 0, 0, 0)
sage: swap((1,1,0,0,1,1,0,0),2)
Traceback (most recent call last):
...
ValueError: There is no such covering move.