# $$\nu$$-Tamari lattice#

A class of the $$\nu$$-Tamari lattice, see [PRV2017] for details.

These lattices depend on one parameter $$\nu$$ where $$\nu$$ is a path of North and East steps.

The elements are nu-Dyck paths which are weakly above $$\nu$$.

To use the provided functionality, you should import $$\nu$$-Tamari lattices by typing:

sage: from sage.combinat.nu_tamari_lattice import NuTamariLattice


Then,

sage: NuTamariLattice([1,1,1,0,0,1,1,0])
Finite lattice containing 6 elements
sage: NuTamariLattice([0,0,0,1,1,0,0,1])
Finite lattice containing 40 elements


The classical Tamari lattices and the Generalized Tamari lattices are special cases of this construction and are also available with this poset:

sage: NuTamariLattice([1,0,1,0,1,0])
Finite lattice containing 5 elements

sage: NuTamariLattice([1,0,0,1,0,0,1,0,0])
Finite lattice containing 12 elements

sage.combinat.nu_tamari_lattice.NuTamariLattice(nu)#

Return the $$\nu$$-Tamari lattice.

INPUT:

• $$\nu$$ – a list of 0s and 1s or a string of 0s and 1s.

OUTPUT:

a finite lattice

The elements of the lattice are nu-Dyck paths weakly above $$\nu$$.

The usual Tamari lattice is the special case where $$\nu = (NE)^h$$ where $$h$$ is the height.

Other special cases give the $$m$$-Tamari lattices studied in [BMFPR].

EXAMPLES:

sage: from sage.combinat.nu_tamari_lattice import NuTamariLattice
sage: NuTamariLattice([1,0,1,0,0,1,0])
Finite lattice containing 7 elements
sage: NuTamariLattice([1,0,1,0,1,0])
Finite lattice containing 5 elements
sage: NuTamariLattice([1,0,1,0,1,0,1,0])
Finite lattice containing 14 elements
sage: NuTamariLattice([1,0,1,0,1,0,0,0,1])
Finite lattice containing 24 elements