Proper caterpillars are distinguished by their symmetric chromatic function

TL;DR

This paper proves that proper caterpillar trees are uniquely identified by their symmetric chromatic functions, confirming the Stanley conjecture for this class by linking caterpillars to integer compositions.

Contribution

It establishes a correspondence between proper caterpillars and integer compositions, proving these trees are distinguished by their symmetric chromatic functions if and only if they are isomorphic.

Findings

Proper caterpillars are uniquely identified by their symmetric chromatic functions.

A correspondence between caterpillars and integer compositions is established.

The Stanley conjecture is confirmed for proper caterpillar trees.

Abstract

This paper deals with the so-called Stanley conjecture, which asks whether they are non-isomorphic trees with the same symmetric function generalization of the chromatic polynomial. By establishing a correspondence between caterpillars trees and integer compositions, we prove that caterpillars in a large class (we call trees in this class proper) have the same symmetric chromatic function generalization of the chromatic polynomial if and only if they are isomorphic.