An integer sequence and standard monomials

Abstract

For an (oriented) graph [Formula: see text] on the vertex set [Formula: see text] (rooted at [Formula: see text]), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal [Formula: see text] in the polynomial ring [Formula: see text] over a field [Formula: see text] such that the number of standard monomials of [Formula: see text] equals the number of (oriented) spanning trees of [Formula: see text] and hence, [Formula: see text], where [Formula: see text] is the truncated Laplace matrix of [Formula: see text]. The standard monomials of [Formula: see text] correspond bijectively to the [Formula: see text]-parking functions. In this paper, we study a monomial ideal [Formula: see text] in [Formula: see text] having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal [Formula: see text] is the cellular resolution supported on a subcomplex of the first barycentric subdivision [Formula: see text] of an [Formula: see text] simplex [Formula: see text]. The integer sequence [Formula: see text] has many interesting properties. In particular, we obtain a formula, [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], similar to [Formula: see text].