Shelling of links and star clusters in edgewise subdivision of a simplex

TL;DR

This paper explores the combinatorial and topological structure of edgewise subdivisions of simplices, introducing new concepts like the faithful initial part statistic and providing explicit formulas for associated invariants.

Contribution

It characterizes the links of vertices in edgewise triangulations using partitions, introduces a new permutation statistic, and provides explicit shellings and formulas for the complexes' h-vectors.

Findings

Links are encoded by partitions of k

Introduces the faithful initial part permutation statistic

Provides explicit shellings and formulas for h-vectors

Abstract

We show that the combinatorial types of the links of the vertices in the edgewise triangulation $T_{k,q}$ of a $(k-1)$-simplex are encoded by the partitions of $k$. Each of these complexes is isomorphic to a subcomplex of the barycentric subdivision of the boundary of a $(k-1)$-simplex, and the containment relations among them are described by a new poset on the set of partitions of $k$. We compute the $h$-vectors of these complexes and determine the number of vertices of $T_{k,q}$ whose links are the same (correspond to the same partition). The combinatorial type of the link of an $(s-1)$-dimensional face of $T_{k,q}$ corresponds to a partition $(\lambda_1,\lambda_2,\ldots,\lambda_s)$ of $k$ into $s$ parts, together with additional partitions of each $\lambda_i$. We also enumerate the combinatorial types of all $m$-dimensional complexes that arise as the links in edgewise triangulations. A new permutation statistic, \textit{the faithful initial part}, is introduced and used to describe the star cluster of a facet of $T_{k,q}$. By examining a specific shelling of this star cluster, we prove that the $i$-th entry of its $h$-vector counts the number of permutations of $[k]$ with exactly $i$ descents, taking into account the faithful initial part as the multiplicity. Finally, we describe a concrete shelling order for $T_{k,q}$, give a combinatorial interpretation of its $h$-vector, and derive an explicit formula for it.