Our purpose in this article is to investigate the order complex of inclusion poset PFn of Borel orbit closures in skew-symmetric matrices. We prove that PFn is an EL-shellable poset and furthermore its order complex triangulates a ball. We investigate (rook-theoretic) combinatorial properties of the rank-generating function of PFn in contrast with the zeta function of the variety of skew-symmet...

We prove a theorem allowing us to find convex-ear decompositions for rankselected subposets of posets that are unions of Boolean sublattices in a coherent fashion. We then apply this theorem to geometric lattices and face posets of shellable complexes, obtaining new inequalities for their h-vectors. Finally, we use the latter decomposition to give a new interpretation to inequalities satisfied ...

We use a concrete shelling order of chessboard complexes ∆n,m for m > 2n − 1 to describe the type of each facet of ∆n,m in this order. Further, we find some recursive relations for h-vector, describe the generating facets of shellable ∆n,m and show that the number of generating facets of ∆n,m is the value of a special Poisson-Charlier polynomial pn(m). Some of these results can be extended to c...

Hyperplane codes are a class of convex codes that arise as the output of a one layer feed-forward neural network. Here we establish several natural properties of nondegenerate hyperplane codes, in terms of the polar complex of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a non-degenerate hyperplane code is shellable and show that all cu...

We define a simplicial complex ∆ W in terms of polygon dissections for every classical reflection group W and positive integer m. For m = 1, ∆ W is isomorphic to the cluster complex corresponding to W , defined in [8]. We enumerate the faces of ∆ W and show that the entries of its h-vector are given by the generalized Narayana numbers N W (i), defined in [3]. We also prove that for any m ≥ 1 th...

We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to...

It is shown that the coset lattice of a finite group has shellable order complex if and only if the group is complemented. Furthermore, the coset lattice is shown to have a Cohen–Macaulay order complex in exactly the same conditions. The group theoretical tools used are relatively elementary, and avoid the classification of finite simple groups and of minimal finite simple groups. © 2006 Elsevi...

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we characterize the antichain cutsets in semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We ...