$\hat{I}\pm$-chromatic symmetric functions

Abstract

In this paper, we introduce the \emph{$\hat{I}\pm$-chromatic symmetric functions} $\ddot{I}{‡}_{\ddot{I}}^{(\hat{I}\pm )}€[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $\hat{I}\pm$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $\hat{I}\pm$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $\hat{I}\pm$-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.

Minor corrections