The $e$-positivity of the chromatic symmetric functions and the inverse Kostka matrix

Shiyun Wang

The $e$-positivity of the chromatic symmetric functions and the inverse Kostka matrix

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arxiv 2210.07567 v2 pith:2FU3WA6B submitted 2022-10-14 math.CO

Shiyun Wang This is my paper · ORCID

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keywords symmetricchromaticfunctionsbouncedyckinversekostkamatrix

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abstract

We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in E\u{g}ecio\u{g}lu-Remmel (1990). We prove that certain coefficients in this expansion are positive. We establish the $e$-positivity of an extended class of chromatic symmetric functions for Dyck paths of bounce number three beyond the "hook-shape" case of Cho-Huh (2019).

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When is the chromatic quasisymmetric function symmetric?
math.CO 2024-12 conditional novelty 8.0 of 10

A graph's chromatic quasisymmetric function is symmetric only under strong constraints; the paper proves connected DAGs with multiple sources/sinks are nonsymmetric and identifies a new symmetric family.

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