Trees, parking functions, syzygies, and deformations of monomial ideals (original) (raw)

Trees, Parking Functions, and Deformations of Monomial Ideals

2018

For a graph G, we define G-parking functions and show that their number is equal to the number of spanning trees of G. We construct a certain monomial ideal and a certain ideal generated by powers of linear forms. The dimension of the quotient of the polynomial ring modulo either of these two ideals equals the number of spanning trees of G. The monomials corresponding to G-parking functions form linear bases in each of these two algebras. Then we investigate 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. We prove several formulas for the Hilbert series.

Trees, parking functions, and standard monomials of skeleton ideals

Australas. J Comb., 2021

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 Mn, a certain monomial ideal in the polynomial ring S = K[x1, . . . , xn] where a set of generators are indexed by the nonempty subsets of [n] = {1, 2, . . . , n}. Motivated by constructions from the theory of chip-firing on graphs we study generalizations of parking functions determined byM n , a subideal ofMn 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 n , denoted stann, contains the usual parking functions and has interesting combinatorial properties in its own right. For general k we show that elements of stann 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 Sn naturally acts on the set ...

Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

arXiv (Cornell University), 2020

Let G be an (oriented) graph on the vertex set V = {0, 1,. .. , n} with root 0. Postnikov and Shapiro associated a monomial ideal M G in the polynomial ring R = K[x 1 ,. .. , x n ] over a field K. The standard monomials of the Artinian quotient R M G correspond bijectively to G-parking functions. A subideal M (k) G of M G generated by subsets of V = V \ {0} of size at most k + 1 is called a k-skeleton ideal of the graph G. Many interesting homological and combinatorial properties of 1-skeleton ideal M (1) G are obtained by Dochtermann for certain classes of simple graph G. A finite sequence P = (p 1 ,. .. , p n) ∈ N n is called a spherical G-parking function if the monomial x P = n i=1 x pi i ∈ M G \ M (n−2) G. Let sPF(G) be the set of all spherical G-parking functions. On counting the number of spherical parking functions of a complete graph K n+1 on V , in two different ways, Dochtermann obtained a new identity for (n−1) n−1. In this paper, a combinatorial description for all multigraded Betti numbers of the k-skeleton ideal M (k) Kn+1 of the complete graph K n+1 on V are given. Also, using DFS burning algorithms of Perkinson-Yang-Yu (for simple graph) and Gaydarov-Hopkins (for multigraph), we give a combinatorial interpretation of spherical G-parking functions for the graph G = K n+1 − {e} obtained from the complete graph K n+1 on deleting an edge e. In particular, we showed that |sPF(K n+1 − {e 0 })| = (n − 1) n−1 for an edge e 0 through the root 0, but |sPF(K n+1 − {e 1 })| = (n − 1) n−3 (n − 2) 2 for an edge e 1 not through the root.

One-skeleta of GGG-parking function ideals: resolutions and standard monomials

arXiv: Combinatorics, 2017

Given a graph GGG, the GGG-parking function ideal MGM_GMG is an artinian monomial ideal in the polynomial ring SSS with the property that a linear basis for S/MGS/M_GS/MG is provided by the set of GGG-parking functions. It follows that the dimension of S/MGS/M_GS/MG is given by the number of spanning trees of GGG, which by the Matrix Tree Theorem is equal to the determinant of the reduced Laplacian of GGG. The ideals MGM_GMG and related algebras were introduced by Postnikov and Shapiro where they studied their Hilbert functions and homological properties. The author and Sanyal showed that a minimal resolution of MGM_GMG can be constructed from the graphical hyperplane arrangement associated to GGG, providing a combinatorial interpretation of the Betti numbers. Motivated by constructions in the theory of chip-firing on graphs, we study certain `skeleton' ideals MG(k)subsetMGM_G^{(k)} \subset M_GMG(k)subsetMG generated by subsets of vertices of GGG of size at most k+1k+1k+1. Here we focus our attention on the case k=1k=1k=1, the ...

Monomial and toric ideals associated to Ferrers graphs

Transactions of the American Mathematical Society, 2008

Each partition λ = (λ1, λ2, . . . , λn) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

Monomial Ideals of Weighted Oriented Graphs

The Electronic Journal of Combinatorics

Let I=I(D)I=I(D)I=I(D) be the edge ideal of a weighted oriented graph DDD whose underlying graph is GGG. We determine the irredundant irreducible decomposition of III. Also, we characterize the associated primes and the unmixed property of III. Furthermore, we give a combinatorial characterization for the unmixed property of III, when GGG is bipartite, GGG is a graph with whiskers or GGG is a cycle. Finally, we study the Cohen–Macaulay property of III.

Symbolic powers of monomial ideals and vertex cover algebras

Advances in Mathematics, 2007

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

The Symmetric Algebra for Certain Monomial Curves 1

arXiv: Commutative Algebra, 2011

Abstract. Let p≥ 2 and 0 <m 0 <m 1 <...<m p be a sequence of positive integers such that they forma minimal arithmetic sequence. Let ℘denote the defining ideal of the monomial curve C in A p+1K , defined bythe parametrization X i = T m i for i∈ [0,p]. Let Rdenote the polynomial ring K[X 1 ,...,X p ,X 0 ]. In thisarticle, we construct a minimal Gro¨bner basis for the symmetric algebra for such curves, as an R-moduleand what is interesting is that the proof does not require any S-polynomial computation. Keywords: Monomial Curves, Gro¨bner Basis, Symmetric Algebra.Mathematics Subject Classification 2000 : 13P10,13A30. 1. Notation Let Ndenote the set ofnon-negative integers andthe symbols a,b,d,i,i ′ ,j,j ′ ,l,l ′ ,m,n,p,q,sdenotenon-negative integers. For our convenience we define [a,b] = {i| a≤ i≤ b} ,ǫ(i,j) =i+j if i+j<pp if i+j≥ pand τ(i,j) =0 if i+j<pp if i+j≥ p 2. Introduction A class of rings, collectively designated Blowup Algebras , appear in many constructi...

The line graph of a tree and its edge ideal

2021

We describe all the trees with the property that the corresponding edge ideal of their line graph has a linear resolution. As a consequence, we give a complete characterization of those trees T for which the line graph L(T ) is cochordal. We compute also the second Betti number of the edge ideal of L(T ) and we determine the number of cycles in L(T ). As a consequence, we obtain also the first Zagreb index of a graph. For edge ideals of line graphs of caterpillar graphs we determine the Krull dimension, the Castelnuovo-Mumford regularity, and the projective dimension under some additional assumption on the degrees of the cutpoints.

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

Journal of Algebraic Combinatorics, 2007

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405-425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs. Keywords Hypergraphs • Chordal graphs • Monomial ideals • Graded resolutions • Regularity Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.