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

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 $({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_s)$ 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.