Extended chromatic symmetric functions and equality of ribbon Schur functions

Abstract

We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specialises 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 generalise to composition of graphs. We then apply our generalisation to obtain infinitely many families of weighted graphs whose members have equal extended chromatic symmetric functions.