Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials

Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that unicellular LLT polynomials have similar properties to graded chromatic symmetric functions. In this paper, we link together the unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties introduced by Ayzenberg-Buchstaber. We prove their palindromicity from topological viewpoint. We also show that modules of a symmetric group generated by f…

正则半单Hessenberg簇的双流形几何与单细胞LLT多项式

Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

本文研究正则半单Hessenberg簇双流形的几何，证明二者通过显式爆破与纤维丛映射关联，取上同调后直接证明模律，确立LLT型Shareshian-Wachs猜想。

LLT多项式与Hecke代数迹

LLT Polynomials and Hecke Algebra Traces

该研究揭示单细胞LLT多项式系数与Hecke代数迹的关联，给出其用传统迹基、归纳法及Kazhdan-Lusztig R-多项式的表达形式，推进相关代数组合研究。

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。

LLT cumulants and graph coloring

The purpose of this note is to introduce a new family of quasi-symmetric functions called LLT cumulants and discuss its properties. We define LLT cumulants using the algebraic framework for conditional cumulants and we prove that the Macdonald cumulant has an explicit positive expansion in terms of LLT cumulants of ribbon shapes, generalizing the classical decomposition of Macdonald polynomials. We also find a natural combinatorial interpretation of the LLT cumulant of a given directed graph as a weighted generating function of colorings of its subgraphs. We use this graph theoretical framewor…

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…

A symmetric function of increasing forests

For an indifference graph $G$ we define a symmetric function of increasing spanning forests of $G$. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular LLT polynomials. As a consequence we give a combinatorial interpretation of the coefficients of the LLT polynomial in the elementary basis (up to a factor of a power of $(q-1)$), strengthening the description given by Alexandersson and Sulzgruber.

Chromatic symmetric functions from the modular law

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced by Guay-Paquet. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain $q$-hit numbers (up to a factor). Finally, we discuss the logarithmic conca…

$P$-划分与$p$-正性

$P$-partitions and $p$-positivity

利用$\alpha$-单峰集的组合学，本文建立了拟对称函数理论中基本基展开与反向$P$-划分生成函数正展开两个新结果，并应用于多种多项式与函数，改进文献中的正性结论。

色函数、区间序与递增森林

Chromatic functions, interval orders and increasing forests

本研究将色拟对称函数从单位区间序推广到一般区间序，证明其为递增生成森林相关拟对称函数的加权和，并推广相关恒等式与猜想。