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

TL;DR

The paper analyzes the edgewise subdivision $T_{k,q}$ of the $(k-1)$-simplex, showing that vertex links are encoded by partitions of $k$ and that a new partition poset $<_c$ governs containment among link complexes. It develops a framework linking links to products of chains, and proves shellability with a detailed interpretation of the $h$-vector via permutation descents and ascents; it introduces the faithful initial part statistic to study the star cluster of facets and derives an explicit shelling and a closed-form formula for $h_i(T_{k,q})$. The results yield a unified combinatorial description of links, stars, and shellings in edgewise triangulations, with connections to partitions, product posets, and known subdivisions such as the barycentric subdivision and Newton polytopes. Collectively, these contributions provide both structural insights into $T_{k,q}$ and concrete enumerative tools for its combinatorial topology.

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 $(λ_1,λ_2,\ldots,λ_s)$ of $k$ into $s$ parts, together with additional partitions of each $λ_i$. 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.

Figures (2)

• Figure 1: The poset $P_{4,2,1}$ describes the link of $v=(0,0,1,1,2,q)\in W_{7,q}$
• Figure 2: The poset $P_F$ for $F=\{\mathbf{v}^{(1)},\mathbf{v}^{(2)},\mathbf{v}^{(3)}\}$ in $T_{8,9}$

Theorems & Definitions (57)

• Proposition 1
• proof
• Example 1: The $h$-vector of $Sd(\partial \Delta^{k-1})$
• Proposition 2
• proof
• Definition 3
• Corollary 4
• Theorem 5
• proof
• Proposition 6