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

We investigate some combinatorial properties of the links and the star clusters of the faces in a concrete triangulation $T_{k,q}$ (the edgewise subdivision) of a $(k-1)$-simplex. We show that the link of a vertex in $T_{k,q}$ is the reduced order complexes of the product of chains and determine the number of combinatorially different links of all vertices of $T_{k,q}$. The combinatorial type of the link of an $s$-dimensional face of $T_{k,q}$ corresponds to the partition $(\lambda_1,\lambda_2,\ldots,\lambda_s)$ of $k$ into $s$ parts and further partitions of each $\lambda_i$. We introduce a new permutation statistic (the faithful initial part) and use it to describe the facets of the star cluster of a facet of $T_{k,q}$. Using a concrete shelling we show that the $i$-th entry of $h$-vector of the star cluster counts the number of permutations from $\mathbb{S}_k$ with exactly $i$ descents, taking into account the faithful initial part as the multiplicity. Finally, we describe a concrete shelling order for $T_{k,q}$, obtain a combinatorial interpretation and find a concrete formula for its $h$-vector.