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

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 Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given k, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet-Tarry-Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism.