Enumeration of certain subsets of uprooted trees and spherical parking functions

Abstract

Spherical $G$-parking functions are a distinguished subset of standard monomials, arising from the skeleton ideals of the $G$-parking function ideal. Explicit enumeration formulas for spherical $G$-parking functions are known only for a few classes of graphs. In this paper, we consider a family of graphs $G_{\ell}$ ($1\leq \ell \leq n-2$), obtained from the complete graph $K_{n+1}$ by deleting the $\ell$ edges joining vertex $1$ to the vertices in $F_{\ell}= \{n-\ell+1, \ldots, n\}$. The uprooted spanning trees of $G_{\ell}-\{0\}$ correspond to the set $\mathcal{U}_n^{1\not\sim F_{\ell}}$ of uprooted trees with vertex set $[n]$ in which vertex $1$ is not adjacent to any vertex in $F_{\ell}$, and we establish that $|\mathcal{U}_n^{1\not\sim F_{\ell}}| = (n-1)^{n-\ell-2}(n-2)^{\ell}(n-\ell-1)$. We derive this formula combinatorially and independently recover it as an application of the matrix tree theorem, obtaining some combinatorial identities as consequences. Finally, we determine the number of spherical $G_{\ell}$-parking functions as $|\mathrm{sPF}(G_{\ell})| = (n-1)^{n-3}(n-\ell-1)^2$.