Trees, parking functions, and standard monomials of skeleton ideals

TL;DR

This paper explores generalized parking functions through algebraic and combinatorial methods, connecting them to trees, group actions, and graph invariants, and introduces new formulas and interpretations for their enumeration.

Contribution

It introduces a new class of generalized parking functions via subideals of monomial ideals, providing formulas for their counts, group action orbits, and combinatorial interpretations related to trees and graph invariants.

Findings

Standard monomials form generalized parking functions with combinatorial properties.

Derived formulas for counting generalized parking functions and their orbits under symmetric group actions.

Connected parking functions to tree structures and graph Laplacians, revealing new combinatorial identities.

Abstract

Parking functions are a widely studied class of combinatorial objects, with connections to several branches of mathematics. On the algebraic side, parking functions can be identified with the standard monomials of $M_n$, a certain monomial ideal in the polynomial ring $S = {\mathbb K}[x_1, \dots, x_n]$ where a set of generators are indexed by the nonempty subsets of $[n] = \{1,2,\dots,n\}$. Motivated by constructions from the theory of chip-firing on graphs we study generalizations of parking functions determined by $M^{(k)}_n$, a subideal of $M_n$ obtained by allowing only generators corresponding to subsets of $[n]$ of size at most $k$. For each $k$ the set of standard monomials of $M^{(k)}_n$, denoted $\text{stan}_n^k$, contains the usual parking functions and has interesting combinatorial properties in its own right. For general $k$ we show that elements of $\text{stan}_n^k$ can be recovered as certain vector-parking functions, which in turn leads to a formula for their count via results of Yan. The symmetric group $S_n$ naturally acts on the set $\text{stan}_n^k$ and we also obtain a formula for the number of orbits under this action. For the case of $k = n-2$ we study combinatorial interpretations of $\text{stan}_n^{n-2}$ and relate them to properties of uprooted trees in terms of root degree and surface inversions. As a corollary we obtain a combinatorial identity for $n^n$ involving Catalan numbers, reminiscent of a result of Benjamin and Juhnke. For the case of $k = 1$ we observe that the number of elements $\text{stan}_n^1$ is given by the determinant of the reduced `signless' Laplacian, which provides a weighted count for $|\text{stan}_n^1|$ in terms generalized spanning trees known as `spanning TU-subgraphs'. Our constructions naturally generalize to arbitrary graphs and lead to a number of open questions.