Enumeration of certain subsets of uprooted trees and spherical parking functions

Spherical $G$-parking functions are a distinguished subset of standard monomials, arising from the skeleton ideals of the $G$-parking function ideal. Explicit spherical $G$-parking function enumeration formulas are known only in a few classes of graphs. In this paper, we consider a family of graphs $\text{Gl}$ ($1\le \mathrm{\ell }\le n-2$), obtained from the complete bipartite $K_{n+1}$ by deleting the $\mathrm{\ell }$ edges joining vertex $1$ to the vertices in $F_{\mathrm{\ell }}=\{n-\mathrm{\ell }+1,\dots ,n\}$. The uprooted spanning trees of $\text{Gl}-\{0\}$ are counted by the set $\text{UnFl}$ of uprooted trees with the vertex set $[n]$ in which vertex $1$ is not adjacent to any vertex in $F_{\mathrm{\ell }}$, and we establish that $\mathrm{\mid }\text{UnFl}\mathrm{\mid }=(n-1{)}^{n-\mathrm{\ell }-2}(n-2{)}^{\mathrm{\ell }}(n-\mathrm{\ell }-1)\mathrm{.}$ 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 $\text{Gl}$-parking functions as $\mathrm{\mid }\text{SPF}(\text{Gl})\mathrm{\mid }=(n-1{)}^{n-3}(n-\mathrm{\ell }-1{)}^2\mathrm{.}$