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 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().

sage.combinat.tamari_lattices.GeneralizedTamariLattice(a, b, m=1)

Returns 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]

Bousquet-Melou, E. Fusy, L.-F. Preville Ratelle. “The number of intervals in the m-Tamari lattices”. arXiv:1106.1498

sage.combinat.tamari_lattices.TamariLattice(n)

Returns 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: from sage.combinat.tamari_lattices import TamariLattice
sage: TamariLattice(3)
Finite lattice containing 5 elements

sage.combinat.tamari_lattices.paths_in_triangle(i, j, a, b)

Returns 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)]

sage.combinat.tamari_lattices.swap(p, i, m=1)

Performs 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.