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

Chromatic symmetric functions of Dyck paths and q-rook theory

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-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. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the $q$-CSFs i…

使用相连车放置的着色拟对称多项式的Hall--Littlewood展开

Hall--Littlewood expansions of chromatic quasisymmetric functions using linked rook placements

本文利用相连车放置描述了自然单位区间序产生的着色拟对称函数的Hall--Littlewood展开系数，并基于Carlsson--Mellit关系进一步给出了单细胞LLT多项式相应展开系数的组合描述。

Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive

Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a positive expansion for this polynomial in the basis of fundamental slide polynomials using recent work of Assaf-Bergeron on flagged $(P,ρ)$-partitions. We then derive the known expansion for the chromatic quasisymmetric function of Dyck graphs in terms of Gessel's fundamental basis by taking a backstable limit of our expansion.

$α$-色对称函数

$α$-chromatic symmetric functions

本文引入带实参数$α$的$α$-色对称函数扩展经典色对称函数，给出基于$α$-二项式基和$α$-下降阶乘基的正组合公式，并建立与$q$-车理论的联系，解决Garsia和Remmel提出的$q$-命中问题。

Gessel polynomials, rooks, and extended Linial arrangements

We study a family of polynomials associated with ascent-descent statistics on labeled rooted plane k-ary trees introduced by Gessel, from a rook-theoretic perspective. We generalize the excedance statistic on permutations to maximal nonattacking rook placements on certain rectangular boards by decomposing them into boards of staircase shape. We then relate the number of maximal nonattacking rook placements on certain skew boards to the number of regions in extended Linial arrangements by establishing a relation between the factorial polynomial of those boards to the characteristic polynomial o…

LLT polynomials, chromatic quasisymmetric functions and graphs with cycles

We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as vertical strip --- in particular, unicellular LLT polynomials. We show that there are parallel phenomena regarding $e$-positivity of these two families of polynomials. In particular, we give several examples where the LLT polynomials behave like a "mirror image" of the chromatic quasisymmetric counterpart. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials. This circular extensions of LLT polynomials…