初等对称函数基下竖直带LLT多项式的组合展开

A Refinement of the Murnaghan-Nakayama Rule by Descents for Border Strip Tableaux

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a $k$-th primitive root of unity yields the number of border strip tableaux with all strips of size $k$, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we intr…

循环筛分、斜麦克唐纳多项式与Schur正性

Cyclic sieving, skew Macdonald polynomials and Schur positivity

该研究证明了分拆部分为n的倍数时，特殊化非对称麦克唐纳多项式对应的填充集合满足循环筛分现象，引入其斜版本并证明Schur正性，还得到一类新LLT多项式的组合Schur展开。

色对称拟对称函数的一种幂零实现

A unipotent realization of the chromatic quasisymmetric function

本文通过有限一般线性群 $\mathrm{GL}_{n}(\mathbb{F}_{q})$ 的复特征标理论实现了色对称拟对称函数和垂直条带 LLT 多项式，提出了一种基于幂零上三角群归纳和 Hopf 代数的计算方法，并揭示了其与 Hessenberg 簇的联系。

$P$-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。

The canonical strip, I

We introduce a canonical strip hypothesis for Fano varieties. We show that the canonical strip hypothesis for a Fano variety implies that the zeros of the Hilbert polynomial of embedded Calabi--Yau and general type hypersurfaces are located on a vertical line. This extends, in particular, Villegas's `polynomial RH' for intersections in projective spaces to the case of CY and general type hyperplane sections in Grassmannians. We state a few conjectures on the Ehrhart polynomials of certain fan polytopes.

三行正方形格条上的自回避多边形

Self-avoiding polygons on a three-row square-lattice strip

该研究给出三行正方形格条上自回避多边形的计数闭式公式，推导相关子类型的二项式和表示，通过两种表示得到恒等式的新几何证明，并建立其与序列A007909的关联。

Toward the Hanani-Tutte Theorem for Clustered Graphs

The weak variant of Hanani-Tutte theorem says that a graph is planar, if it can be drawn in the plane so that every pair of edges cross an even number of times. Moreover, we can turn such a drawing into an embedding without changing the order in which edges leave the vertices. We prove a generalization of the weak Hanani-Tutte theorem that also easily implies the monotone variant of the weak Hanani-Tutte theorem by Pach and Tóth. Thus, our result can be thought of as a common generalization of these two neat results. In other words, we prove the weak Hanani-Tutte theorem for strip clustered gr…