Trees, parking functions, syzygies, and deformations of monomial ideals

Abstract

Abstract. For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

“…Combining Theorem with [, Corollary 1.5] yields our last result. In the case when one of the hypergraphs induced by the bipartite graph is in fact a graph, this easily follows from a formula of Merino via the connection noted in between parking functions and the chip firing game (a.k.a. abelian sandpile model).…”

“…Shapiro and the second author defined so‐called parking functions for an arbitrary directed graph . These objects have a natural enumerator , which is a one‐variable polynomial with positive integer coefficients.…”

“…Observations by Murakami and the first author concerning generalised parking functions for the dual of a plane bipartite graph yield that their natural enumerator also coincides with the interior polynomial of (see Corollary ).…”

Between Shi and Ish

“…, x n ) ∈ F n q | ∀1 ≤ i < j ≤ n , x i = x j as injective labelings in F q of the elements of [n] [1,14]. For instance, for n = 5 and q = 13, the labeling represented below corresponds to x := (2,9,3,11,12) ∈ S 5 13 . Note that if (x 1 , .…”

Parking Functions on Directed Graphs and Some Directed Trees

“…Parking functions have seen many generalizations: G-parking functions [11], u-parking functions [9], parking sequences [4], rational parking functions [1], and those defined on tree-shaped parking lots [2,10]. In this paper, we extend the "drivers searching for a parking spot" analogy from the trees in [10] to general digraphs and give a description that generalizes the set definition of the classical parking functions in Section 2.…”