An instance of umbral methods in representation theory: the parking function module

关于素停车函数、卢卡希夫路径和拟对称函数的统计学

On statistics of prime parking functions, Łukasiewicz paths, and quasisymmetric functions

本文研究了素停车函数的统计特性，推导了位移平均值公式，并建立了Schur函数与拟对称函数之间的联系。

超图上的停车函数与筹码发射

Parking functions and chip-firing on hypergraphs

本文将图上的停车函数与筹码发射概念扩展至超图，定义了超图停车函数，证明其极大序列由无环定向刻画，且可由$q$-根生成树计数，并揭示了其与超图筹码发射超稳定构型的等价关系。

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

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…

Tutte Short Exact Sequences of Graphs

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…