A generalization of Foata's fundamental transformation and its applications to the right-quantum algebra

通过Foata基本变换将循环限制转化为网格模式

Turning cycle restrictions into mesh patterns via Foata's fundamental transformation

本文通过Foata基本变换，将排列中相邻q-循环的数量映射为两种网格模式的出现次数之和，并以此解决了第一作者先前论文中的猜想3.14。

Transformation à la Foata for special kinds of descents and excedances

A pure excedance in a permutation $π=π_1π_2\ldots π_n$ is a position $i<π_i$ such that there is no $j<i$ with $i\leq π_j<π_i$. We present a one-to-one correspondence on the symmetric group that transports pure excedances to descents of special kind. As a byproduct, we prove that the popularity of pure excedances equals those of pure descents on permutations, while their distributions are different.

秉承“基本变换”精神的两个双射

Two bijections in the spirit of the transformation fondamentale

本文用双射方法证明了两个联系循环与线性统计量的等分布结果，其思路秉承Foata“基本变换”的精神。

Left-Right Pairs and Complex Forests of Infinite Rooted Binary Trees

Let $\mathcal{D}_0:= \{x + iy \ \vert x, y >0\}$, and let $(L, R)$ be a pair of Möbius transformations corresponding to $\mathrm{SL}_2(\mathbb{N}_0)$ matrices such that $R(\mathcal{D}_0)$ and $L(\mathcal{D}_0)$ are disjoint. Given such a pair (called a left-right pair), we can construct a directed graph $\mathcal{F}(L, R)$ with vertices $\mathcal{D}_0$ and edges $\{(z, R(z))\}_{z \in \mathcal{D}_0} \cup \{(z, L(z))\}_{z \in \mathcal{D}_0}$, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of $\mathrm{SL}_2(\mathbb{N}_…