The Partitionability Conjecture

Duval, Art M.; Klivans, Caroline J.; Martin, Jeremy L.

doi:10.1090/noti1475

Cited by 3 publications

(2 citation statements)

References 2 publications

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“…We note the similarity between the resolution of this conjecture and the Partitionability Conjecture (see, e.g., [6,14]). A complex ∆ is partitionable if its face poset can be written as the disjoint union of boolean intervals whose maximal faces are the facets of ∆.…”

Section: Boolean Treesmentioning

confidence: 55%

Resolving Stanley's conjecture on $k$-fold acyclic complexes

Doolittle¹,

Goeckner²

2021

Combinatorial Theory

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In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank 1 boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.

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Section: Boolean Treesmentioning

confidence: 55%

Resolving Stanley's conjecture on $k$-fold acyclic complexes

Doolittle¹,

Goeckner²

2021

Combinatorial Theory

View full text Add to dashboard Cite

show abstract

“…We note the similarity between the resolution of this conjecture and the Partitionability Conjecture (see, e.g., [8,17]). A complex ∆ is partitionable if its face poset can be written as the disjoint union of boolean intervals whose maximal faces are the facets of ∆.…”

Section: Boolean Treesmentioning

confidence: 55%

Resolving Stanley's conjecture on $k$-fold acyclic complexes

Doolittle¹,

Goeckner²

2018

Preprint

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In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank 1 boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove both a weaker version and a special case of the original conjecture.

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Partitioning the projective plane and the dunce hat

Santamaría-Galvis

2022

European Journal of Combinatorics

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The Partitionability Conjecture

Cited by 3 publications

References 2 publications

Resolving Stanley's conjecture on $k$-fold acyclic complexes

Resolving Stanley's conjecture on $k$-fold acyclic complexes

Resolving Stanley's conjecture on $k$-fold acyclic complexes

Partitioning the projective plane and the dunce hat

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