Friezes, weak friezes, and T-paths

非交换带状图案及其行列式、弱带状图案的非交换洛朗现象与带状图案拼接

Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing

本文研究Coxeter带状图案的非交换推广，推广了行列式公式、表述洛朗现象的$T$-路径公式及拼接结果，以非交换版弱带状图案为核心工具。

Weak friezes and frieze pattern determinants

Frieze patterns have been introduced by Coxeter in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Canakci and Jorgensen, are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any friez…

循环求和现象与蛇形图案

The Cyclic Sieving Phenomenon and frieze patterns

本文研究了循环求和现象在多边形划分和三角剖分中的应用，并通过双射关系扩展了蛇形图案与 Dyck 路径的联系。

$SL_k$-Tilings and Paths in $\mathbb{Z}^k$

An $SL_k$-tiling is a bi-infinite array of integers having all adjacent $k\times k$ minors equal to one and all adjacent $(k+1)\times (k+1)$ minors equal to zero. Introduced and studied by Bergeron and Reutenauer, $SL_k$-tilings generalize the notion of Conway-Coxeter frieze patterns in the case $k=2$. In a recent paper, Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and $SL_2$-tilings. We extend this result to higher $k$ by constructing a bijection between $SL_k$-tilings and certain pairs of bi-infinite strips of vectors in $\mathbb{Z}^k$ called pat…

A fast new algorithm for weak graph regularity

We provide a deterministic algorithm that finds, in $ε^{-O(1)} n^2$ time, an $ε$-regular Frieze-Kannan partition of a graph on $n$ vertices. The algorithm outputs an approximation of a given graph as a weighted sum of $ε^{-O(1)}$ many complete bipartite graphs. As a corollary, we give a deterministic algorithm for estimating the number of copies of $H$ in an $n$-vertex graph $G$ up to an additive error of at most $εn^{v(H)}$, in time $ε^{-O_H(1)}n^2$.

An Expansion Formula for Decorated Super-Teichmüller Spaces

Motivated by the definition of super-Teichmüller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichmüller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $λ$-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $λ$-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$-path formulas for type $A$ cluster algebras. We further connect our formulas to the super-friezes of Morie…