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arXiv · 2407.06965

$\alpha$-chromatic symmetric functions

Jim Haglund·Jaeseong Oh·Meesue Yoo
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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.

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Jim Haglund, Jaeseong Oh, Meesue Yoo. 2024-07-09. $\alpha$-chromatic symmetric functions. https://arxiv.org/abs/2407.06965

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