Combinatorial nature of ground state vector of O(1) loop model

The Loop Murnaghan-Nakayama Rule

We give a combinatorial proof of a natural generalization of the Murnaghan-Nakayama rule to loop Schur functions. We also define shifted loop Schur functions and prove that they satisfy a similar relation.

Combinatorial Aspects of the Distribution of Rough Objects

The inverse problem of general rough sets, considered by the present author in some of her earlier papers, in one of its manifestations is essentially the question of when an agent's view about crisp and non crisp objects over a set of objects has a rough evolution. In this research the nature of the problem is examined from number-theoretic and combinatorial perspectives under very few assumptions about the nature of data and some necessary conditions are proved.

Quantized cohomological Hall algebra of the $d$-loop quiver revisited

Let $Λ$ be the set of partitions of length $\geq 0$. We introduce an $\mathbb{N}$-graded algebra $\mathbb{A}_q^d(Λ)$ associated to $Λ$, which can be viewed as a quantization of the algebra of partitions defined by Reineke. The multiplication of $\mathbb{A}^d_q(Λ)$ has some kind of quasi-commutativity, and the associativity comes from combinatorial properties of certain polynomials appeared in the quantized cohomological Hall algebra $\mathcal{H}^d_q$ of the $d$-loop quiver. It turns out that $\mathbb{A}^d_q(Λ)$ is isomorphic to $\mathcal{H}^d_q$, thus can be viewed as a combinatorial realizati…

Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain

The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations.

关于Kazhdan--Lusztig多项式组合不变性的一个注记

A note on Combinatorial Invariance of Kazhdan--Lusztig polynomials

本文引入惊人超立方体分解及双捷径概念，提出蕴含对称群Kazhdan--Lusztig多项式组合不变性猜想的组合性猜想，附录还在余初等区间下证明了双超立方体的类似猜想。

A few words about maps

In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the f…

圆柱形Hitomezashi图案

Cylindrical Hitomezashi Patterns

本研究给出了圆柱形hitomezashi图案上非零同调类环的组合描述，类比Pete通过同高度Dyck路径对双射的方法，同时完善了Ren和Zhang关于环形hitomezashi环的部分研究结果。

朴素离散同伦论中的映射纤维、环路与悬垂图

Mapping fiber, loop and suspension graphs in naive discrete homotopy theory

本文通过定义并研究映射纤维图及其性质，在技术性条件下构造出朴素离散同伦论中的 Puppe 序列类比，以揭示 A-同伦论与普通同伦论的异同。