\(\nu\)-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
See also
For more detailed information see NuTamariLattice()
. For more
information on the standard Tamari lattice see
sage.combinat.tamari_lattices.TamariLattice()
,
sage.combinat.tamari_lattices.GeneralizedTamariLattice()
- 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