Chromatic classical symmetric functions

TL;DR

This paper classifies when skew Schur functions are positive linear combinations of power sum functions and characterizes which scalar multiples of classical symmetric functions can be realized as chromatic symmetric functions of graphs.

Contribution

It provides a complete classification of skew Schur functions in terms of power sum positivity and identifies specific classical symmetric functions realizable as chromatic functions.

Findings

Skew Schur functions are positive linear combinations of power sums under specific conditions.

Only scalar multiples of elementary symmetric functions can be realized as chromatic symmetric functions of certain graphs.

Certain unions of complete graphs correspond to these realizable elementary symmetric functions.

Abstract

In this note we classify when a skew Schur function is a positive linear combination of power sum symmetric functions. We then use this to determine precisely when any scalar multiple of a skew Schur function is the chromatic symmetric function of some graph. From here we are able to prove that of the classical bases for symmetric functions only certain scalar multiples of the elementary symmetric functions can be realised as the chromatic symmetric function of some graph, namely a particular union of complete graphs.