2024-04-15

## 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.

## References

- [ACSZ20] JosÃ© Aliste-Prieto, Logan Crew, Sophie Spirkl and JosÃ© Zamora. A vertex-weighted Tutte symmetric function, and constructing graphs with equal chromatic symmetric function. arXiv e-prints, 2020.
- [AS19b] Per Alexandersson and Robin Sulzgruber. P-partitions and P-positivity. International Mathematics Research Notices, July 2019.
- [AWW20] Farid Aliniaeifard, Victor Wang and Stephanie van Willigenburg. Extended chromatic symmetric functions and equality of ribbon Schur functions. arXiv e-prints, 2020.
- [CS19] Logan Crew and Sophie Spirkl. A deletion-contraction relation for the chromatic symmetric function. arXiv e-prints, 2019.
- [CS20] Logan Crew and Sophie Spirkl. A complete multipartite basis for the chromatic symmetric function. arXiv e-prints, 2020.