Craig Desjardins
In this note, we find a monomization of a certain power ideal associated to a directed graph. This power ideal has been studied in several settings. The combinatorial method described here extends earlier work of other, and will work on several other types of power ideals, as will appear in later work.
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The $G$-parking function ideal $M_G$ of a directed multigraph $G$ is a monomial ideal which encodes some of the combinatorial information of $G$. It is an initial ideal of the toppling ideal $I_G$, a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for $M_G$ was given by Postnikov and Shaprio in the case when $G$ is saturated, i.\,e., whenever there is at least one edge $(u,v)$ for every ordered pair of distinct vertices $u$ and $v$. They also raised the problem of an explicit description o…
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 sh…
Given a graph $G$, the $G$-parking function ideal $M_G$ is an artinian monomial ideal in the polynomial ring $S$ with the property that a linear basis for $S/M_G$ is provided by the set of $G$-parking functions. It follows that the dimension of $S/M_G$ is given by the number of spanning trees of $G$, which by the Matrix Tree Theorem is equal to the determinant of the reduced Laplacian of $G$. The ideals $M_G$ 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…
Let $G$ be an (oriented) graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro associated a monomial ideal $\mathcal{M}_G$ in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$. A subideal $\mathcal{M}_G^{(k)}$ of $\mathcal{M}_G$ generated by subsets of $\widetilde{V}=V\setminus \{0\}$ 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 $\mathcal{M}_G^{(1)}$ are obtained by Dochtermann for certain classes of simple graph $G$. A finite…
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 configuration. In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of th…
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 $M_n$, a certain monomial ideal in the polynomial ring $S = {\mathbb K}[x_1, \dots, x_n]$ where a set of generators are indexed by the nonempty subsets of $[n] = \{1,2,\dots,n\}$. Motivated by constructions from the theory of chip-firing on graphs we study generalizations of parking functions determined by $M^{(k)}_n$, a subideal of $M_n$ obtained by allowing only generators correspo…
Enumeration of certain subsets of uprooted trees and spherical parking functions
本文研究从完全二部图K_{n+1}删除特定边得到的图族，枚举了无根树子集并推导出计数公式，进而得到球形停车函数个数。
We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and contains the descent algebra as a left ideal.
On Parking Functions and The Tower of Hanoi
本文建立长度为 $n$、位移为一的停车函数与 $n+1$ 盘 $n+1$ 柱汉诺塔理想状态之间的双射，并指出二者均由 Lah 数计数。
We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. The $G$-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied $G$-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and an edge deletion $G \setminus e$ ($e$ is a non-bridge). We obtain purely combinatorial consequences of Tutte short…
We present a new proof of the monomial case of Wilmes' conjecture, which gives a formula for the coarsely-graded Betti numbers of the G-parking function ideal in terms of maximal parking functions of contractions of G. Our proof is via poset topology and relies on a theorem of Gasharov, Peeva, and Welker that connects the Betti numbers of a monomial ideal to the topology of its lcm-lattice.
The Priority Lattice
本文引入了优先级格，证明其最大链由停车函数标记，主理想的最大链由部分停车函数标记，并计算了其格级数和特征多项式。
We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n+1)^{n-1} in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schroder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to some family of di- or tri-algebras occuring in the works of Loday and Ronco. Moreover, the fundament…
Restricted Chip-Firing: Toric Toppling Ideals, Picard Groups, and Cellular Resolutions
本文研究了pargraphs上的芯片翻转游戏所产生的一类群和理想，探讨了其拓扑理想性质，并构建了格罗布纳基、最小细胞自由解析及皮卡群的自由性条件。
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