Elliptic enumeration of nonintersecting lattice paths

A doubly-refined enumeration of alternating sign matrices and descending plane partitions

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of -1's in A and the number of 0's to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of n's in D. Here, the result is generalized to include a fourth st…

加权象限游走的枚举：代数性与D-有限性的判据

Enumeration of weighted quadrant walks: criteria for algebraicity and D-finiteness

本文提出基于椭圆函数理论的统一方法，为象限内加权游走生成函数的代数性与D-有限性判定提供统一证明。

Broken bracelets, Molien series, paraffin wax and an elliptic curve of conductor 48

This paper introduces the concept of necklace binomial coefficients motivated by the enumeration of a special type of sequences. Several properties of these coefficients are described, including a connection between their roots and an elliptic curve. Further links are given to a physical model from quantum mechanical supersymmetry as well as properties of alkane molecules in chemistry.

椭圆曲线、Riordan阵列与格路径

Elliptic Curves, Riordan arrays and Lattice Paths

本文建立椭圆曲线与格路径之间的对应关系，证明两者分别通过椭圆可除序列和Hankel变换关联到本质上相同的Somos 4序列，并用Riordan阵列完成路径计数。

Combinatorial families of multilabelled increasing trees and hook-length formulas

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree familie…

Hurwitz Theory of Elliptic Orbifolds, I

An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for the full modular group $SL_2(\mathbb{Z})$. They later generalized this theorem to the enumeration of branched covers of a pillowcase, i.e. the quotient of an elliptic curve by the elliptic involution, proving quasi-modularity for $Γ_1(2)$. We generalize their work to the quotient of an elliptic curve by cyclic groups of orders $N=3$, $4$, $6$, proving quasi-m…

受限雅可比排列

Restricted Jacobi permutations

本文系统研究雅可比排列中的模式避免问题，完整枚举了避免长度3模式的雅可比排列，并借助二叉树子族、Dyck路径双射与生成函数技巧，对单一模式限制情形给出了关于上升数、从左到右极小值数及末位字母的精细枚举。

Winding of simple walks on the square lattice

A method is described to count simple diagonal walks on $\mathbb{Z}^2$ with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the generating functions of which are encoded in a commuting set of Hilbert space operators. The general enumeration problem is then solved by obtaining an explicit eigenvalue decomposition of these operators involving elliptic functions. By further restricting the intermediate winding angles of the walks to some open interval, the method can be use…

象限中离散热核展开里的对数项

Logarithmic terms in discrete heat kernel expansions in the quadrant

本文提出基于 Jacobi theta 函数与 Jacobi 变换的生成函数方法，计算四分之一平面格点游走计数的完整渐近展开，并首次在无限群模型中发现对数项。