-chromatic symmetric functions

In this paper, we introduce the -chromatic symmetric functions ^ () _;q, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter. We present positive combinatorial formulas and provide explicit interpretations. Notably, we show an explicit monomial expansion in terms of the -binomial basis and an expansion into certain chromatic symmetric functions in terms of the -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.