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Parking functions are a widely studied class of combinatorial objects, withconnections to several branches of mathematics. On the algebraic side, parkingfunctions can be identified with the standard monomials of $M_n$, a certainmonomial ideal in the polynomial ring $S = {\mathbb K}[x_1, \dots, x_n]$ wherea set of generators are indexed by the nonempty subsets of $[n] =\{1,2,\dots,n\}$. Motivated by constructions from the theory of chip-firing ongraphs we study generalizations of parking functions determined by $M^{(k)}_n$,a subideal of $M_n$ obtained by allowing only generators corresponding tosubsets of $[n]$ of size at most $k$. For each $k$ the set of standardmonomials of $M^{(k)}_n$, denoted $\text{stan}_n^k$, contains the usual parkingfunctions and has interesting combinatorial properties in its own right. Forgeneral $k$ we show that elements of $\text{stan}_n^k$ can be recovered ascertain vector-parking functions, which in turn leads to a formula for theircount 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 underthis action. For the case of $k = n-2$ we study combinatorial interpretationsof $\text{stan}_n^{n-2}$ and relate them to properties of uprooted trees interms of root degree and surface inversions. As a corollary we obtain acombinatorial identity for $n^n$ involving Catalan numbers, reminiscent of aresult of Benjamin and Juhnke. For the case of $k = 1$ we observe that thenumber 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'. Ourconstructions naturally generalize to arbitrary graphs and lead to a number ofopen questions.

Trees, parking functions, and standard monomials of skeleton ideals