SU（2）_k的Bratteli图中路的计数

有限域上支持在斜Young图和Rothe图补上的矩阵计数

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

我们考虑的问题，找到一个有限域上的矩阵的数量与一定的秩和支持，避免了一个子集的项目。这些矩阵是具有受限位置的置换的q模拟（即，车放置）。对于一般的元素集合，这些矩阵的个数不是q中的多项式（Stembridge 98）;然而，当元素集合是一个杨氏图时，这些数，直到q-1的幂，都是具有非负系数的多项式（Haglund 98）。本文给出了这类数是q中的多项式或系数为非负整数的多项式的若干条件。我们扩展Haglund的结果，以补充斜杨图，我们将此结果的情况下，当一组条目是Rothe图的置换。特别地，我们给出了一个置换的Rothe图是斜Young图的补图直到行和列重排的充要条件。最后，我们给出了连接可逆矩阵的支撑避免了Rothe图和庞加莱多项式的强Bruhat阶。

动力系统组合近似：分离图方法

Combinatorial approximations of dynamical systems: a separated graph approach

本文提出用分离图方法构建Bratteli-like分离图，用于近似动力系统的动态结构，通过四个典型例子展示其应用，并讨论如何从分离图中读取最小动力系统。

Counting fixed points free vector fields on $\mathbb{B}^{2}$

The number of diagrams of stationary points free vector fields in the 2-disk $\mathbb{B}^{2}$ is counted in the article. It is shown that the number of such diagrams with $2k$ exceptional points on the boundary $\mathbb{S}^{1}$ equals $3^{k-2}(C_{k}+2C_{k-1})$, where $C_{k}$ is the corresponding Catalan number. An algorithm for finding all such diagrams is discussed.

色多项式、Colered Jones函数与q-二项式计数

Chromatic Polynomial, Colered Jones Function and q-Binomial Counting

在一般弦图类中定义了图的q-色函数，给出了它的一些性质，并给出了一些公式。然后，我们把q-色函数与纽结的色琼斯函数联系起来。这导致了一个奇怪的表达的有色琼斯函数的结图作为一个“缺陷色运营商”适用于幂级数的系数是线性组合的弦图构建从“流”的减少结图。

Counting non-crossing permutations on surfaces of any genus

Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leadi…

A short conceptual proof of Narayana's path-counting formula

We deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.

Catalan triangles and tied arc diagrams

The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.

矩阵积分与缠结和链接的计数

Matrix Integrals and the Counting of Tangles and Links

利用矩阵模型技术计算平面费曼图，再现了Sundberg和Thistlethwaite最近关于交替缠结和连接的计算结果。

Counting tropical rational space curves with cross-ratio constraints

This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio condit…

广义O（n）四次矩阵模型及其在计算缠结和链环中的应用

The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links

交替缠结计数的交叉数，外部腿和连接组件的数量在这里提出了一个统一的框架，使用量子场论的方法适用于一个矩阵模型的彩色链接。利用矩阵模型的重正化方案，消除了与图的拓扑等价有关的超算，导出了相应的重正化方程.对一些特殊情况进行了详细的研究，并得到了精确的解决.

矩阵积分与虚缠结和虚链环的产生和计数

Matrix Integrals and the Generation and Counting of Virtual Tangles and Links

虚链路是经典链路的推广，可以用嵌入“加厚”曲面$Σ\times I$中的链路来表示，该曲面是亏格$h$的黎曼曲面与区间的乘积。在本文中，我们表明，虚拟交替链接和缠结自然与$1/N^2$展开的积分在$N\times N$复杂的矩阵。我们建议，这是足够的计数这些图模普通（平面）苍蝇的等价类。为了验证这一假设，我们使用的算法编码相应的费曼图的排列，生成虚拟图多达6个交叉点，并计算各种不变量。在此假设下，我们利用已知的矩阵积分结果，得到亏格为1到5直到10阶的虚交错缠结的生成函数（即\ 10真正的十字路口）。对于$n$的渐近行为，大的数目的链接和缠结的属$h$和与$n$的交叉也计算了$h=1,2,3$和一般$h$。