2024-06-17

## 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 [CS20a], 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 [AWvW21], the authors study weighted paths with equal extended chromatic symmetric functions.

In [AWvW21], 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

- [AS19b] Per Alexandersson and Robin Sulzgruber. P-partitions and P-positivity. International Mathematics Research Notices, 2021(14):10848–10907, July 2019.
- [AWvW21] Farid Aliniaeifard, Victor Wang and Stephanie van Willigenburg. Extended chromatic symmetric functions and equality of ribbon Schur functions. Advances in Applied Mathematics, 128:102189, July 2021.
- [CS19] Logan Crew and Sophie Spirkl. A deletion-contraction relation for the chromatic symmetric function. arXiv e-prints, 2019.
- [CS20a] Logan Crew and Sophie Spirkl. A complete multipartite basis for the chromatic symmetric function. arXiv e-prints, 2020.