Combinatorial and Probabilistic Formulae for Divided Symmetrization

TL;DR

This paper explores combinatorial and probabilistic interpretations of divided symmetrization for functions associated with trees, providing new insights and generalizations for specific cases like paths and connecting to volume computations of polytopes.

Contribution

It offers a combinatorial interpretation for divided symmetrization on trees and a probabilistic game model for paths, extending known results and suggesting broader generalizations.

Findings

Combinatorial interpretation for monomials on trees

Probabilistic game interpretation for path graphs

Connection to volume computations of polytopes

Abstract

Divided symmetrization of a function $f(x_1,\dots,x_n)$ is symmetrization of the ratio $$DS_G(f)=\frac{f(x_1,\dots,x_n)}{\prod (x_i-x_j)},$$ where the product is taken over the set of edges of some graph $G$. We concentrate on the case when $G$ is a tree and $f$ is a polynomial of degree $n-1$, in this case $DS_G(f)$ is a constant function. We give a combinatorial interpretation of the divided symmetrization of monomials for general trees and probabilistic game interpretation for a tree which is a path. In particular, this implies a result by Postnikov originally proved by computing volumes of special polytopes, and suggests its generalization.