Decomposable clutters and a generalization of Simon's conjecture

Bigdeli, Mina; Pour, Ali Akbar Yazdan; Zaare-Nahandi, Rashid

doi:10.1016/j.jalgebra.2019.03.037

Cited by 12 publications

(9 citation statements)

References 21 publications

(39 reference statements)

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“…In [2] the authors consider a notion of a decomposable d-clutter that generalizes chordal graphs in a different way, and conjecture that such clutters admit a simplicial ridge. As spelled out in [2] a positive answer to this question would imply Simon's conjecture. As far as we know this conjecture is open even for the case of d = 3.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

Culbertson

Dochtermann

Guralnik

et al. 2020

Electron. J. Combin.

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Section: Resultsmentioning

confidence: 99%

“…The Björner-Eriksson result establishes the k = 2 case of Simon's conjecture by considering U 3 n , the uniform matroid of rank 3. In [2] Simon's conjecture was established for k = n − 3 (a simpler proof was provided independently in [10] based on results from [7]). Here we prove that much more is true.…”

Section: Introductionmentioning

confidence: 99%

Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

Culbertson

Dochtermann

Guralnik

et al. 2020

Electron. J. Combin.

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“…The face poset of (∆, F ) is pictured below. A partitioning of this poset is given by the intervals [23, 23], [3,34], [4,24].…”

Section: Intermediate Constructionsmentioning

confidence: 99%

Partition and Cohen-Macaulay Extenders

Doolittle¹,

Goeckner²,

Lazar³

2019

Preprint

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If a pure simplicial complex is partitionable, then its h-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex ∆, we construct a complex Γ ⊇ ∆ of the same dimension such that both Γ and the relative complex (Γ, ∆) are partitionable. This allows us to rewrite the h-vector of any pure simplicial complex as the difference of two h-vectors of partitionable complexes, giving an analogous interpretation of the h-vector of a non-partitionable complex.By contrast, for a given complex ∆ it is not always possible to find a complex Γ such that both Γ and (Γ, ∆) are Cohen-Macaulay. We characterize when this is possible, and we show that the construction of such a Γ in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.

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“…Recently Bigdeli et al [4] and Dochtermann et al [10,Conjecture 4.8] [11] have proposed a new approach based on the graph-theoretic notion of chordality, which allowed them to establish Simon's conjecture for d ≥ n − 3, and even to show that shellability is equivalent to extendable shellability for d-complexes with up to d + 3 vertices [11].…”

Section: Introductionmentioning

confidence: 99%