Chromatic symmetric functions of Dyck paths and q -rook theory
Chromatic symmetric functions of Dyck paths and q -rook theory
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Dyck路径的色对称函数和q-rook理论
DOI：
10.1016/j.ejc.2022.103595
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发表时间：
2023
影响因子：
1
通讯作者：
Panova, Greta
中科院分区：
文献类型：
--
作者：
Colmenarejo, Laura;Morales, Alejandro H.;Panova, Greta
Stanley的Dyck路的色对称函数（CSF）及其Shareshian-Wachs q-模拟与Hessenberg簇、对角调和函数和LLT多项式有着重要的联系。在所谓的阿贝尔情形中，它们也与Stanley and Stembridge（1993）和Guay-Paquet（2013）的结果中的非攻击车的位置有奇怪的关系。对于q-模拟，Abreu和Nigro（2021）和Guay-Paquet（私人通信）使用q-命中数推广了这些结果。在我们的主要成果是一个新的证明Guay-Paquet的优雅的身份表示的q-CSF在CSF基础上与q-命中系数。我们进一步证明了它等价于Abreu-Nigro恒等式在初等对称函数中展开q-CSF。在我们的工作过程中，我们建立了这些展开式中的q-击中数与最初假设的Garsia-Remmel q-击中数相差q的某些幂。我们证明了这些q-hit数的新身份，并建立了三个不同的变体之间的联系。
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, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley and Stembridge (1993) and Guay-Paquet (2013). For the q-analogue, these results have been generalized by Abreu and Nigro (2021) and Guay-Paquet (private communication), using q-hit numbers. 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. In the course of our work we establish that the q-hit numbers in these expansions differ from the originally assumed Garsia–Remmel q-hit numbers by certain powers of q. We prove new identities for these q-hit numbers, and establish connections between the three different variants.