Generalized Tamari lattices¶
These lattices depend on three parameters \(a\), \(b\) and \(m\), where \(a\) and \(b\) are coprime positive integers and \(m\) is a nonnegative integer.
The elements are Dyck paths
in the \((a \times b)\)-rectangle. The order relation depends on \(m\).
To use the provided functionality, you should import Generalized Tamari lattices by typing:
sage: from sage.combinat.tamari_lattices import GeneralizedTamariLattice
Then,
sage: GeneralizedTamariLattice(3,2)
Finite lattice containing 2 elements
sage: GeneralizedTamariLattice(4,3)
Finite lattice containing 5 elements
The classical Tamari lattices are special cases of this construction and are also available directly using the catalogue of posets, as follows:
sage: posets.TamariLattice(3)
Finite lattice containing 5 elements
See also
For more detailed information see TamariLattice()
,
GeneralizedTamariLattice()
.
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sage.combinat.tamari_lattices.
GeneralizedTamariLattice
(a, b, m=1)¶ Return the \((a,b)\)-Tamari lattice of parameter \(m\).
INPUT:
- \(a\) and \(b\) coprime integers with \(a \geq b\)
- \(m\) a nonnegative integer such that \(a \geq b \times m\)
OUTPUT:
- a finite lattice (the lattice property is only conjectural in general)
The elements of the lattice are
Dyck paths
in the \((a \times b)\)-rectangle.The parameter \(m\) (slope) is used only to define the covering relations. When the slope \(m\) is \(0\), two paths are comparable if and only if one is always above the other.
The usual Tamari lattice of index \(b\) is the special case \(a=b+1\) and \(m=1\).
Other special cases give the \(m\)-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] M. Bousquet-Melou, E. Fusy, L.-F. Preville Ratelle. The number of intervals in the m-Tamari lattices. Arxiv 1106.1498
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sage.combinat.tamari_lattices.
TamariLattice
(n)¶ Return the \(n\)-th Tamari lattice.
INPUT:
- \(n\) a nonnegative integer
OUTPUT:
- a finite lattice
The elements of the lattice are
Dyck paths
in the \((n+1 \times n)\)-rectangle.See Tamari lattice for mathematical background.
EXAMPLES:
sage: posets.TamariLattice(3) Finite lattice containing 5 elements
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sage.combinat.tamari_lattices.
paths_in_triangle
(i, j, a, b)¶ Return all Dyck paths from \((0,0)\) to \((i,j)\) in the \((a \times b)\)-rectangle.
This means that at each step of the path, one has \(a y \geq b x\).
A path is represented by a sequence of \(0\) and \(1\), where \(0\) is an horizontal step \((1,0)\) and \(1\) is a vertical step \((0,1)\).
INPUT:
- \(a\) and \(b\) coprime integers with \(a \geq b\)
- \(i\) and \(j\) nonnegative integers with \(1 \geq \frac{j}{b} \geq \frac{bi}{a} \geq 0\)
OUTPUT:
- a list of paths
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)]
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sage.combinat.tamari_lattices.
swap
(p, i, m=1)¶ Perform a covering move in the \((a,b)\)-Tamari lattice of parameter \(m\).
The letter at position \(i\) in \(p\) must be a \(0\), followed by at least one \(1\).
INPUT:
- \(p\) a Dyck path in the \((a \times b)\)-rectangle
- \(i\) an integer between \(0\) and \(a+b-1\)
OUTPUT:
- a Dyck path in the \((a \times b)\)-rectangle
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.