The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier (1992). Björner, Paffenholz, Sjöstrand and Ziegler (2005) generalize this construction to obtain a Bier poset Bier(P, I) from any bounded poset P and any proper ideal I ⊆ P . They show shellability of Bier(P, I) for the case P = Bn, the boolean lattice, and thereby obtain ‘many shellable spheres’ in the sense ...

In 1992, Thomas Bier introduced a surprisingly simply way to construct a large number of simplicial spheres. He proved that, for any simplicial complex ∆ on the vertex set V with ∆ 6= 2 , the deleted join of ∆ with its Alexander dual ∆∨ is a combinatorial sphere. In this paper, we extend Bier’s construction to multicomplexes, and study their combinatorial and algebraic properties. We show that ...

In this article we prove that the poset of m-divisible noncrossing partitions is EL-shellable for every wellgenerated complex reflection group. This was an open problem for type G(d, d, n) and for the exceptional types, for which a proof is given case-by-case. Résumé. Dans cet article nous prouvons que l’ensemble ordonné des partitions non-croisées m-divisibles est ELépluchable (“EL-shellable”)...

The tropical semiring (R,min,+) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of n points on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of n × d matrices of tropical rank two, a simplici...

Each positive rational number x > 0 can be written uniquely as x = a/(b − a) for coprime positive integers 0 < a < b. We will identify x with the pair (a, b). In this extended abstract we use rational Dyck paths to define for each positive rational x > 0 a simplicial complex Ass(x) = Ass(a, b) called the rational associahedron. It is a pure simplicial complex of dimension a− 2, and its maximal ...

In Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are B-shellable. Our result shows that the whole Oriented Matroid is shellable, too.

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...