In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when polynomial (real) stable, a property often deduced via theory interlacing polynomials. Many open questions on stability pertain to enumeration faces cell complexes. this paper, we relate shellability We first derive sufficient condition ...

We prove that if a simplicial complex ∆ is (nonpure) shellable, then the intersection lattice for the corresponding real coordinate subspace arrangement A∆ is homotopy equivalent to the link of the intersection of all facets of ∆. As a consequence, we show that the singularity link of A∆ is homotopy equivalent to a wedge of spheres. We also show that the complement of A∆ is homotopy equivalent ...

We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion shellability for such complexes. It is shown that q-complexes formed by independent subspaces q-matroid are shellable. Further, we explicitly determine homology corresponding to uniform q-matroids. also outline some partial results concerning determination arbitrary shellable q-complexes.

We give a complete enumeration of combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product S×S and 615 triangulations of the twisted sphere product S×S. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertexminimal non-shellable 3-balls with 9 vertices.

A recent conjecture that appeared in three papers by Bigdeli–Faridi, Dochtermann, and Nikseresht, is every simplicial complex whose clique has shellable Alexander dual, ridge-chordal. This strengthens the long-standing Simon's k-skeleton of simplex extendably shellable, for any k. We show stronger a negative answer, exhibiting an infinite family counterexamples.

We recently introduced a notion of tilings the geometric realization finite simplicial complex and related those to discrete Morse theory R. Forman, especially when they have property be shellable, shared by classical shellable complexes. now observe that every such tiling supports quiver which is acyclic precisely then shelling induces two spectral sequences converge (co)homology complex. Thei...

After [4] the shellability of multicomplexes Γ is given in terms of some special faces of Γ called facets. Here we give a criterion for the shellability in terms of maximal facets. Multigraded pretty clean filtration is the algebraic counterpart of a shellable multicomplex. We give also a criterion for the existence of a multigraded pretty clean filtration.