D-Complete Posets

AUTHORS:

• Stefan Grosser (06-2020): initial implementation

class sage.combinat.posets.d_complete.DCompletePoset(hasse_diagram, elements, category, facade, key)

Bases: FiniteJoinSemilattice

A d-complete poset.

D-complete posets are a class of posets introduced by Proctor in [Proc1999]. It includes common families such as shapes, shifted shapes, and rooted forests. Proctor showed in [PDynk1999] that d-complete posets have decompositions in irreducible posets, and showed in [Proc2014] that d-complete posets admit a hook-length formula (see Wikipedia article Hook_length_formula). A complete proof of the hook-length formula can be found in [KY2019].

EXAMPLES:

Sage

sage: from sage.combinat.posets.poset_examples import Posets
sage: P = Posets.DoubleTailedDiamond(2)
sage: TestSuite(P).run()

Python

>>> from sage.all import *
>>> from sage.combinat.posets.poset_examples import Posets
>>> P = Posets.DoubleTailedDiamond(Integer(2))
>>> TestSuite(P).run()

get_hook(elmt)

Return the hook length of the element elmt.

EXAMPLES:

Sage

sage: from sage.combinat.posets.d_complete import DCompletePoset
sage: P = DCompletePoset(DiGraph({0: [1], 1: [2]}))
sage: P.get_hook(1)
2

Python

>>> from sage.all import *
>>> from sage.combinat.posets.d_complete import DCompletePoset
>>> P = DCompletePoset(DiGraph({Integer(0): [Integer(1)], Integer(1): [Integer(2)]}))
>>> P.get_hook(Integer(1))
2

get_hooks()

Return all the hook lengths as a dictionary.

EXAMPLES:

Sage

sage: from sage.combinat.posets.d_complete import DCompletePoset
sage: P = DCompletePoset(DiGraph({0: [1, 2], 1: [3], 2: [3], 3: []}))
sage: P.get_hooks()
{0: 1, 1: 2, 2: 2, 3: 3}
sage: from sage.combinat.posets.poset_examples import Posets
sage: YDP321 = Posets.YoungDiagramPoset(Partition([3,2,1]))
sage: P = DCompletePoset(YDP321._hasse_diagram.reverse())
sage: P.get_hooks()
{0: 5, 1: 3, 2: 1, 3: 3, 4: 1, 5: 1}

Python

>>> from sage.all import *
>>> from sage.combinat.posets.d_complete import DCompletePoset
>>> P = DCompletePoset(DiGraph({Integer(0): [Integer(1), Integer(2)], Integer(1): [Integer(3)], Integer(2): [Integer(3)], Integer(3): []}))
>>> P.get_hooks()
{0: 1, 1: 2, 2: 2, 3: 3}
>>> from sage.combinat.posets.poset_examples import Posets
>>> YDP321 = Posets.YoungDiagramPoset(Partition([Integer(3),Integer(2),Integer(1)]))
>>> P = DCompletePoset(YDP321._hasse_diagram.reverse())
>>> P.get_hooks()
{0: 5, 1: 3, 2: 1, 3: 3, 4: 1, 5: 1}

hook_product()

Return the hook product for the poset.