α-Chromatic Symmetric Functions

Abstract In this paper, we introduce the $\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.