D-Complete Posets¶
AUTHORS:
Stefan Grosser (06-2020): initial implementation
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class
sage.combinat.posets.d_complete.
DCompletePoset
(hasse_diagram, elements, category, facade, key)¶ Bases:
sage.combinat.posets.lattices.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: from sage.combinat.posets.poset_examples import Posets sage: P = Posets.DoubleTailedDiamond(2) sage: TestSuite(P).run()
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get_hook
(elmt)¶ Return the hook length of the element
elmt
.EXAMPLES:
sage: from sage.combinat.posets.d_complete import DCompletePoset sage: P = DCompletePoset(DiGraph({0: [1], 1: [2]})) sage: P.get_hook(1) 2
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get_hooks
()¶ Return all the hook lengths as a dictionary.
EXAMPLES:
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: P = DCompletePoset(Posets.YoungDiagramPoset(Partition([3,2,1]))._hasse_diagram.reverse()) sage: P.get_hooks() {0: 5, 1: 3, 2: 1, 3: 3, 4: 1, 5: 1}
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hook_product
()¶ Return the hook product for the poset.
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