Extended chromatic symmetric functions and equality of ribbon Schur functions

TL;DR

This paper establishes a broad relation for extended chromatic symmetric functions of weighted graphs, introduces methods for new symmetric function bases, and classifies when weighted paths have equal functions, linking to ribbon Schur functions.

Contribution

It generalizes the classification of equal extended chromatic symmetric functions for weighted paths and introduces a composition operation for graphs.

Findings

Classified when two weighted paths have equal extended chromatic symmetric functions.

Developed two methods to generate new bases for the algebra of symmetric functions.

Identified infinitely many families of weighted graphs with equal extended chromatic symmetric functions.

Abstract

We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) $k$-deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to the classification of equal ribbon Schur functions. This latter classification is dependent on the operation composition of compositions, which we generalize to composition of graphs. We then apply our generalization to obtain infinitely many families of weighted graphs whose members have equal extended chromatic symmetric functions.