Interpreting the two variable Distance enumerator of the Shi hyperplane arrangement

Abstract

We give an interpretation of the coefficients of the two variable refinement $D_{\Sh_n}(q,t)$ of the distance enumerator of the Shi hyperplane arrangement $\Sh_n$ in $n$ dimensions. This two variable refinement was defined by Stanley \cite{stan-rota} for the general $r$-extended Shi hyperplane arrangements. We give an interpretation when $r=1$. We define three natural three-dimensional partitions of the number $(n+1)^{n-1}$. The first arises from parking functions of length $n$, the second from special posets on $n$ vertices defined by Athanasiadis and the third from spanning trees on $n+1$ vertices. We call the three partitions as the parking partition, the tree-poset partition and the spanning-tree partition respectively. We show that one of the parts of the parking partition is identical to the number of edge-labelled trees with label set $\{1,2,...,n\}$ on $n+1$ unlabelled vertices. We prove that the parking partition majorises the tree-poset partition and conjecture that the spanning-tree partition also majorises the tree-poset partition.