On trees with the same restricted U-polynomial and the Prouhet-Tarry-Escott problem

TL;DR

This paper investigates the problem of whether non-isomorphic trees can share the same restricted U-polynomial, and constructs such trees using solutions to the Prouhet-Tarry-Escott problem, revealing new classes distinguished by the U-polynomial.

Contribution

It introduces a method to construct non-isomorphic trees with identical restricted U-polynomials using the Prouhet-Tarry-Escott problem, advancing understanding of tree invariants.

Findings

Constructed non-isomorphic trees with same U_k-polynomial for any k

Identified new classes of trees distinguished by the U-polynomial

Linked the Prouhet-Tarry-Escott problem to tree polynomial invariants

Abstract

This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same $U$-polynomial (or, equivalently, the same chromatic symmetric function). We consider the $U_k$-polynomial, which is a restricted version of $U$-polynomial, and construct with the help of solutions of the Prouhet-Tarry-Escott problem, non-isomorphic trees with the same $U_k$-polynomial for any given $k$. By doing so, we also find a new class of trees that are distinguished by the $U$-polynomial up to isomorphism.