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

@article{Postnikov2003TreesPF, title={Trees, parking functions, syzygies, and deformations of monomial ideals}, author={Alexander Postnikov and Boris Z. Shapiro}, journal={Transactions of the American Mathematical Society}, year={2003}, volume={356}, pages={3109-3142} }

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… Expand

#### 153 Citations

Combinatorics and geometry of power ideals

- Mathematics
- 2008

We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay… Expand

On power ideals of transversal matroids and their "parking functions".

- Mathematics
- 2018

To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the… Expand

Monomization of Power Ideals and Generalized Parking Functions

- 2014

A power ideal is an ideal in a polynomial ring generated by powers of homogeneous linear forms. Power ideals arise in many areas of mathematics, including the study of zonotopes, approximation… Expand

G-PARKING FUNCTIONS AND MINIMAL FREE RESOLUTIONS OF POWERS OF LINEAR FORMS

- 2013

For a graph G, Postnikov-Shapiro [PS04] construct two ideals IG and JG. IG is a monomial ideal and JG is generated by powers of linear forms. They proved the equality of the Hilbert series and… Expand

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

- Mathematics
- 2016

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay… Expand

On the multigraded Hilbert and Poincaré–Betti series and the Golod property of monomial rings

- Mathematics
- 2005

Abstract In this paper we study the multigraded Hilbert and Poincare–Betti series of A = S / a , where S is the ring of polynomials in n indeterminates divided by the monomial ideal a . There is a… Expand

Ordinary and Generalized Circulation Algebras for Regular Matroids

- Mathematics
- 2018

Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (xe)e∈E, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of… Expand

On the Golod property of Stanley–Reisner rings

- Mathematics
- 2007

Abstract Recently in [M. Jollenbeck, On the multigraded Hilbert and Poincare series of monomial rings, J. Pure Appl. Algebra 207 (2) (2006) 261–298] the second author made a conjecture about the… Expand

Hierarchical zonotopal power ideals

- Computer Science, Mathematics
- Eur. J. Comb.
- 2012

This work unifies and generalizes results by Ardila and Postnikov on power ideals and by Holtz and Ron, and Holtz et al. on (hierarchical) zonotopal algebra. Expand

Divisors on Graphs, Connected Flags, and Syzygies

- Mathematics
- 2012

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Grobner theory. We give an explicit description of a minimal Grobner basis… Expand

#### References

SHOWING 1-10 OF 49 REFERENCES

Cellular resolutions of monomial modules

- Mathematics
- 1997

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results… Expand

The LCM-lattice in monomial resolutions

- Mathematics
- 1999

Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: • constructing… Expand

Linear systems on a special rational surface

- Mathematics
- 2003

We study the Hilbert series of a family of ideals J_\phi generated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints… Expand

Generic and Cogeneric Monomial Ideals

- Mathematics, Computer Science
- J. Symb. Comput.
- 2000

This paper states that reverse lexicographic initial ideals of generic lattice ideals are generic, and the Cohen–Macaulay property implies shellability for both the Scarf complex and the Stanley–Reisner complex. Expand

Algebras of Curvature Forms on Homogeneous Manifolds

- Mathematics
- 1999

Let C(X) be the algebra generated by the curvature two-forms of standard holomorphic hermitian line bundles over the complex homogeneous manifold X = G=B. The cohomology ring of X is a quotient of… Expand

Infinite dimensional Lie algebras: Frontmatter

- Mathematics, Physics
- 1990

Introduction Notational conventions 1. Basic definitions 2. The invariant bilinear form and the generalized casimir operator 3. Integrable representations of Kac-Moody algebras and the weyl group 4.… Expand

Monomial Resolutions

- Mathematics
- 1996

Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal free… Expand

A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2002

In the second subdivision of Πn(x), the chambers are indexed in a natural way by rooted binary trees with n+1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron. Expand

Polynomial ideals for sandpiles and their Gröbner bases

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2002

A polynomial ideal encoding topplings in the abelian sandpile model on a graph is introduced. A Grbner basis of this ideal is interpreted combinatorially in terms of well-connected subgraphs. This… Expand

On the Enumeration of Generalized Parking Functions

- Mathematics
- 1999

Let x = (x1, x2, . . . , xn) ∈ Nn . Define a x-parking function to be a sequence (a1, a2, . . . , an) of positive integers whose non-decreasing rearrangement b1 ≤ b2 ≤ · · · ≤ bn satisfies bi ≤ x1 +… Expand