# The symmetric functions catalog

An overview of symmetric functions and related topics

2022-12-30

## Extended chromatic symmetric functions

L. Crew and S. Spirkl [CS19] introduce a vertex-weighted version of the chromatic symmetric functions. This has the advantage that it fulfills a deletion-contraction relation. Furthermore, this family of polynomials is also $\omega \powerSum$-positive. This can easily be see from [AS19b].

In [CS20], the authors examine a new basis, the complete multipartite basis, which is closely related to the extended chromatic symmetric functions.

The deletion-contraction relation does not extend to the $q$-weighed version with ascends.

In [AWW20], the authors study weighted paths with equal extended chromatic symmetric functions.

In [ACSZ20], the authors consider a Tutte-symmetric extension of the vertex-weighted chromatic symmetric functions. This generalizes both Tutte symmetric functions and the chromatic symmetric functions. They provide a spanning-tree formula for these, which then provide a new spanning-tree formula for the chromatic symmetric functions.