Proper caterpillars are distinguished by their symmetric chromatic function

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.

A new polynomial on compositions of integers: on distinguishing caterpillars from their symmetric chromatic function

In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factorization, namely that reversing irreducible factors does not change L, and conjecture that is the only way of generating such compositions. Finally, we find a sufficient condition for two caterpillars have a different symmetric function generalization of the chromatic polynomial, and use this condition to show that if our conjecture were to hold, then the symmetric function generalization of the chromatic polynomial distinguishes among a large class of caterpillars.