Quasisymmetric functions distinguishing trees

TL;DR

This paper explores whether certain quasisymmetric functions can uniquely identify trees and posets with tree-like structures, extending previous results and proposing new conjectures in algebraic combinatorics.

Contribution

It generalizes existing results on rooted trees and formulates conjectures about the distinguishing power of chromatic quasisymmetric functions for directed trees.

Findings

Proves the case of rooted trees for the conjecture

Proposes a new conjecture relating quasisymmetric functions and directed trees

Extends previous results to more general tree structures

Abstract

A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the $P$-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.