标题： 染色对称函数与Dyck路径及q-棋子理论（扩展摘要） 显示英文标题

标题： Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)

摘要： Dyck路径的色对称函数（CSF）由Stanley提出，以及其Shareshian-Wachs$q$-类似物与Hessenberg流形、对角调和函数和LLT多项式有重要联系。在所谓的阿贝尔Dyck路径的情况下，它们也通过Stanley-Stembridge（1993）和Guay-Paquet（2013）的结果与非攻击性车的放置有趣地相关。对于$q$-类似物，这些结果已由Abreu-Nigro（2020）和Guay-Paquet（私人通信）通过$q$-击中数进行推广，这些是Garsia和Remmel引入的版本的变体。我们主要结果之一是对Guay-Paquet优雅恒等式的新的证明，该恒等式以$q$-击中系数在CSF基中表达$q$-CSFs。我们进一步展示了它与Abreu-Nigro将$q$-CSF展开为初等对称函数的恒等式的等价性。这是FPSAC扩展摘要版本。完整版本在ArXiv：2104.07599。 显示英文摘要

摘要： The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs $q$-analogue have important connections to Hessenberg varieties, diagonal harmonics, and LLT polynomials. In the case of, so-called, abelian Dyck paths they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the $q$-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using $q$-hit numbers, which are a variant of the ones introduced by Garsia and Remmel. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the $q$-CSFs in a CSF basis with $q$-hit coefficients. We further show its equivalence to the Abreu-Nigro identity expanding the $q$-CSF in the elementary symmetric functions. This is the FPSAC extended abstract version. The full version is at ArXiv: 2104.07599.