n次相邻对换后的逆序数期望

Limiting behavior of the distance of a random walk

This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here, we study three new examples. It is trivial that the distance for random walk on the hypercube is smooth and is given by one simple formula. In the case of random adjacent transpositions, we find that there is no phase transition even though the distance has different scalings in three different regimes. In the case of a random 3-regular graph, there is a ph…

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。

交错群中的循环排序

Circular sorting in the alternating group

该研究探讨交错群中用3-循环排序偶置换的最大步数，针对不同模4余值的$n$确定或分析了排序次数，证明较大值会无限次出现。

Expected length of a product of random reflections

We present a simple formula for the expected number of inversions in a permutation of size $n$ obtained by applying $t$ random (not necessarily adjacent) transpositions to the identity permutation. More general, for any finite irreducible Coxeter group belonging to one of the infinite families (type A, B, D, and I), an exact expression is obtained for the expected length of a product of $t$ random reflections.

The Local Limit of Random Sorting Networks

A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of $an$ for $a\in[0,1]$ as $n\to\infty$. Here time is scaled by a factor of $1/n$ and space is not scaled. The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on…

随机子词与管道梦

Random Subwords and Pipe Dreams

本文研究由相邻对换词经随机字母删除得到的随机排列，给出一大类起始词逆序数期望的渐近公式，并在特定词情形证明常数与 $n^{3/2}$ 增长，解决相关猜想。

Consecutive patterns in inversion sequences II: avoiding patterns of relations

Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the classical (non-consecutive) case, and later by Auli--Elizalde in the consecutive case, where the entries of a pattern are required to occur in adjacent positions. In this paper we continue this investigation by considering {\em consecutive patterns of relations}, in analogy to the work of Martinez--Savage in the classical case. Specifically, given two binary re…

图的反转直径的边数界

Edge-Number Bounds for the Inversion Diameter of Graphs

研究有向图反转图直径的边数界，证明上界为2√|E(G)|，下界为|E(G)|/|V(G)|，并改进二部图的上界。

通过量子电路进行排列的随机采样

Random sampling of permutations through quantum circuits

本文受 Steinhaus-Johnson-Trotter 算法启发，提出经典及量子电路模型下的排列随机采样算法，并应用于双样本随机化检验，同时提出嵌套 corona 乘积图生成模型以支持特定排列集合的采样。

反演图的直径

Diameter of the inversion graph

本文研究标号图反演图的直径，揭示其与星色数、无环色数和有向色数的有界性等价，并给出多类图的上界及 NP 难性。