$\alpha$-chromatic symmetric functions

TL;DR

This paper introduces $oldsymbol{ ext{$oldsymbol{oldsymbol{ ext{ extalpha}}}$-chromatic symmetric functions}}$, extending existing functions with a parameter, providing combinatorial formulas, and linking to $q$-rook theory and the $q$-hit problem.

Contribution

It defines $ ext{$ extalpha$-chromatic symmetric functions}$, offers explicit combinatorial expansions, and connects to $q$-rook theory, including solving the $q$-hit problem.

Findings

Derived positive combinatorial formulas for $ extalpha$-chromatic symmetric functions.

Established explicit monomial and basis expansions.

Linked $ extalpha$-chromatic functions to $q$-rook theory and solved the $q$-hit problem.

Abstract

In this paper, we introduce the \emph{$\alpha$-chromatic symmetric functions} $\chi^{(\alpha)}_\pi[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $\alpha$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $\alpha$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $\alpha$-falling factorial basis. Among various connections with other subjects, we highlight a significant link to $q$-rook theory, including a new solution to the $q$-hit problem posed by Garsia and Remmel in their 1986 paper introducing $q$-rook polynomials.