Unimodality and the reflection principle

A Cheeger inequality for graphs based on a reflection principle

Given a graph with a designated set of boundary vertices, we define a new notion of a Neumann Laplace operator on a graph using a reflection principle. We show that the first eigenvalue of this Neumann graph Laplacian satisfies a Cheeger inequality.

A Reflection Principle for Three Vicious Walkers

We establish a reflection principle for three lattice walkers and use this principle to reduce the enumeration of the configurations of three vicious walkers to that of configurations of two vicious walkers. In the combinatorial treatment of two vicious walkers, we make connections to two-chain watermelons and to the classical ballot problem. Precisely, the reflection principle leads to a bijection between three walks $(L_1, L_2, L_3)$ such that $L_2$ intersects both $L_1$ and $L_3$ and three walks $(L_1, L_2, L_3)$ such that $L_1$ intersects $L_3$. Hence we find a combinatorial interpretation…

避开一个象限的游走与反射原理

Walks avoiding a quadrant and the reflection principle

本文研究避开负象限的小步长平面格点游走计数，解决了允许八个最近邻步的国王模型，证明其生成函数为D-有限级数与代数级数之和，并推进了七个Weyl步集的代数性现象验证。

非相交路径与自由边界菱形镶嵌的反射原理

A reflection principle for nonintersecting paths and lozenge tilings with free boundaries

本文提出一个补充 Okada–Stembridge Pfaffian 公式的新公式，建立非相交路径的反射原理，将自由边界菱形镶嵌计数归结为无自由边界情形，并给出乘积公式、分解定理及行列式公式等应用。

Test Sets for Nonnegativity of Polynomials Invariant under a Finite Reflection Group

A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ of real homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ if and only if it is nonnegative at all points of $S$. We prove that the union of the hyperplanes perpendicular to the elements of a root system $Φ\subseteq \mathbb{R}^n$ is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by $Φ$. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by $-1$. We then characterize subspaces of forms of arb…