Chromatic MacMahon symmetric functions of graphs

TL;DR

This paper introduces a new graph invariant called the chromatic MacMahon symmetric function, which extends the classical chromatic symmetric function to vertex-weighted graphs using MacMahon symmetric functions, and proves its ability to determine detailed vertex subset statistics for trees.

Contribution

It develops a novel invariant for weighted graphs based on MacMahon symmetric functions and proves its sufficiency to determine complex vertex subset data in trees, generalizing previous unweighted results.

Findings

The chromatic MacMahon symmetric function uniquely determines vertex subset statistics in trees.

This invariant extends classical chromatic symmetric functions to weighted graphs.

The result generalizes a conjecture for unweighted graphs, now proven for weighted cases.

Abstract

A MacMahon symmetric function is an invariant of the diagonal action of the symmetric group on power series in multiple alphabets of variables. We introduce an analogue of the chromatic symmetric function for vertex-weighted graphs, taking values in the MacMahon symmetric functions on two sets of variables, recording information about both cardinalities and weights of vertex sets. We prove that the chromatic symmetric MacMahon function of a tree determines the generating function for its vertex subsets by cardinality, weight, and the numbers of internal and external edges. This result generalizes the one for the unweighted case, first conjectured by Crew and proved independently by Aliste-Prieto--Martin--Wagner--Zamora and Liu--Tang.