分层树、停车函数与 Postnikov-Shapiro 代数

The sandpile model on the complete split graph, Motzkin words, and tiered parking functions

We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars,…

Nonhomogeneous parking functions and noncrossing partitions

For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function.

Tiered trees, weights, and q-Eulerian numbers

Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations for their number. In this paper we generalize this construction and define tiered trees by allowing more than two classes of vertices. Tiered trees arise naturally when counting the absolutely indecomposable representations of certain quivers, and also when one enumerates torus orbits on certain homogeneous varieties. We define a notion of weight for tiered…

素图停车函数与阿贝尔沙堆模型的强递归构型

Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model

本文研究了停车函数与阿贝尔沙堆模型（ASM）的对偶性，定义了“不可分解”的素$G$-停车函数，证明其对应ASM的强递归构型，并在轮图、完全图、完全多部图和完全分裂图等图族上验证了该联系。

Multiparking Functions, Graph Searching, and the Tutte Polynomial

A parking function of length n is a sequence (b_1, b_2,..., b_n) of nonnegative integers whose nondecreasing rearrangement (a_1, a_2,...,a_n) has the property that a_i < i for every i. A well-known result about parking functions is that the polynomial P_n(q), which enumerates the complements of parking functions by the sum of their terms, is the generating function for the number of connected graphs by the number of excess edges when evaluated at (1+q). In this paper we extend this result to arbitrary connected graphs G. In general the polynomial that encodes information about subgraphs of G i…

A refinement of Cayley's formula for trees

A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials P_n(a,b,c)= c(a+(n-1)b+c)(2a+(n-2)b+c)...((n-1)a+b+c) which reduce to (n+1)^{n-1} for a=b=c=1. Our study of proper vertices was motivated by A. Postnikov's hook length formula for binary trees (arXiv:math.CO/0507163), which was also proved by W. Y. C. Chen and L. L. M. Yang (arXiv:math.CO/0507163) and generalized by R. R. X. D…