Olivier Bernardi, Alejandro H. Morales, Richard P. Stanley, Rosena R. X. Du
We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of the random permutation of $\{1,2,...,n\}$ obtained as product of two uniformly random $n$-cycles.
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A celebrated analogy between prime factorizations of integers and cycle decompositions of permutations is explored here. Asymptotic formulas characterizing semismooth numbers (possessing at most several large factors) carry over to random permutations. We offer a survey of practical methods for computing relevant probabilities of a bivariate or trivariate flavor.
We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.
We present various results on multiplying cycles in the symmetric group. Our first result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on $n$ symbols as a product of an $n$-cycle and an $n-1$-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths.…
A family F of permutations of the vertices of a hypergraph H is called "pairwise suitable" for H if, for every pair of disjoint edges in H, there exists a permutation in F in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for H is called the "separation dimension" of H and is denoted by π(H). Equivalently, π(H) is the smallest natural number k so that the vertices of H can be embedded in R^k such that any two disjoint edges of H can be separated by a hyperplane normal to one of the axes. We show that the separation dimen…
In this paper, we introduce the rhombic alternative tableaux, whose weight generating functions provide combinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there are two species of particles, one heavy and one light, hopping right and left on a one-dimensional finite lattice with open boundaries. Parameters $α$, $β$, and $q$ describe the hopping probabilities. The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assemblées of permutations. We describe a bijection between the rhombic alternative table…
In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of $n$-cycles whose product has $k$ cycles and has $m$ given elements contained in distinct cycles (or separated) is given by $$ \frac{2 (n-1)! C_m(n+1,k)}{(n+m)(n+1-m)} $$ when $n-k$ is even, where $C_m(n,k)$ is the number of permutations of $n$ elements having $k$ cycles and separating $m$ given elements. As consequences, we obtain the formulas for certain separation probabilities due to Du and Stanley, answering a call of Stanley for simple combinat…
An ordinal pattern for a finite sequence of real numbers is a permutation that records the relative positions in the sequence. For random walks with steps drawn uniformly from $[-1,1]$, we show an ordinal pattern occurs with probability $\frac{|[1,w]|}{2^n n!}$, where $[1,w]$ is a weak order interval in the affine Weyl group $\widetilde{A}_n$. For random walks with steps drawn from a symmetric Laplace distribution, the probability is $\frac{1}{2^n \prod_{j=1}^n \mathrm{lev}(π)_j}$, where $\mathrm{lev}(π)_j$ measures how often $j$ occurs between consecutive values in $π$. Permutations whose con…
Let $i(\infty,k)$ be the limiting proportion, as $n \rightarrow \infty$, of permutations in the symmetric group of degree $n$ that fix a $k$-set. We give an algorithm for computing $i(\infty,k)$ and state the values of $i(\infty,k)$ for $k \le 30$. These values are consistent with a conjecture of Peter Cameron that $i(\infty,k)$ is a decreasing function of $k$.
Connections Between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings
本文研究无指定间隔的组合计数问题，通过构造与受限排列、组合及比特串等价类之间的多个双射，并给出若干情形下的生成函数求法。
For a given permutation $π_n$ in $S_n$, a random permutation graph is formed by including an edge between two vertices $i$ and $j$ if and only if $(i - j) (π_n(i) - π_n (j)) < 0$. In this paper, we study various statistics of random permutation graphs. In particular, the degree of a given node, the number of nodes with a given degree, the number of isolated vertices, and the number of cliques are analyzed. Further, explicit formulas for the probabilities of having a given number of connected components and isolated vertices are obtained.
On the frequency of permutations containing a long cycle
本文给出了有限对称群$S_n$中阶元除$m$的比例$P(n,m)$的一个一般的显式上界，其中对于某个常数$c$，$n-1 \le m \le cn$.这是用来找到的条件概率的下限，$S_n$或$A_n$的元素包含一个$r$ -循环，假定它满足一个方程的形式$x^{rs}=1$其中$s\leq3$。例如，给定$x^n=1$，元素$x$是$n$ -循环的条件概率总是大于2/7，并且如果$n$不除24则大于1/2。我们的研究结果提高了估计这些条件概率在早期的工作中的作者与比尔斯，Leedham-Green和Seress，并有应用程序分析黑盒识别算法的有限对称和交替组。
A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $π(H)$. Equivalently, $π(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane norma…
Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies over all the permutations in S{n}, and epsilon{i} varies over the set {+1, -1}. By the paper "Hultman Numbers and Generalized Commuting Probability in Finite Groups" (2017), The case where all epsilon{i} are +1 led to a close connection to Hultman numbers. In this paper we generalize the results, permitting epsilon{i} to be -1. We describe the s…
A permutation of [n] induces a graph on [n] such that the edges of the graph correspond to inversion pairs of the permutation. This graph is connected if and only if the corresponding permutation is indecomposable. Let s(n,m) denote a permutation chosen uniformly at random among all permutations of [n] with exactly m inversions. Let p(n,m) be the common value for the probabilities that s(n,m) is indecomposable or the corresponding graph is connected. We prove that p(n,m) is non-decreasing with m by constructing a Markov process in which s(n,m+1) is obtained from s(n,m) by increasing one of the…
Let $G$ be a finite group and $π$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}} \] when $π$ varies over all the permutations in $S_{n}$. The probability \[ Pr_π(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}}) \] is identical to $Pr_{1}^ω(G)$, with \[ ω=a_{1}a_{2}...a_{n-1}a_{n}a_{π_{1}}^{-1}a_{π_{2}}^{-1}\cdots a_{π_{n-1}}^{-1}a_{π_{n}}^{-1}, \] as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativit…
The record statistic and forward stability of Schubert products
本文通过排列的记录统计量研究Schubert多项式积的前向稳定性，分析了均匀排列、Grassmannian排列和布尔排列的记录概率，并推导了其渐进行为。
Common models for random graphs, such as Erdős-Rényi and Kronecker graphs, correspond to generating random adjacency matrices where each entry is non-zero based on a large matrix of probabilities. Generating an instance of a random graph based on these models is easy, although inefficient, by flipping biased coins (i.e. sampling binomial random variables) for each possible edge. This process is inefficient because most large graph models correspond to sparse graphs where the vast majority of coin flips will result in no edges. We describe some not-entirely-well-known, but not-entirely-unknown,…
Online Permutation Embedding: Optimal Stopping and Scaling Laws
该研究针对将k元置换嵌入均匀随机变量流的问题，提出最优在线嵌入算法及动态规划方法，发现随机置换、单调置换、极值置换的嵌入速度存在严格层级差异，与离线理论猜想相悖。
Integrable Lattices: Random Matrices and Random Permutations
这些讲座提出了随机矩阵（有限和无限）和随机排列领域的最新发展的调查。这些概率问题提出了矩阵积分（或弗雷德霍姆行列式），它非常自然地出现在对称空间的切空间上的积分，群上的积分，最后是对称空间上的积分。这些讲座的一个重要部分是致力于表明，这些矩阵积分，在适当地添加时间参数，是自然的τ-功能的可积格，如户田，普法夫和Toeplitz格，但也可积偏微分方程的，如KdV方程。这些矩阵积分或弗雷德霍姆行列式也满足Virasoro约束，其与可积方程组合导致原始概率的（部分）微分方程。
We show that the random transposition walk on the symmetric group $S_n$ has cutoff in separation distance at $\frac{1}{2}n \log n$, by constructing a strong stationary time. The construction involves working with cycle types of permutations and some partition combinatorics.
Permutations from Random Walk
本文研究随机游走部分和排序产生的排列，证明其分布与步长分布无关，并推广至一般情形，引入Wiener permuton，给出插值构造，统一扩展了反正弦定律。
Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
本文研究了随机置换、随机不动点自由对合和反向随机不动点自由对合的泊松化系综中递增序列的最大长度的累积概率Pr（L（t）\le 1）.结果表明，对于酉对称、正交对称和辛对称的矩阵系综，这些概率分别等于硬边间隙概率。间隙概率可以写为某些决定点过程的相关性之和。从这些表达式可以证明Pr（L（t）\le 1）在三种情况下的极限形式分别等于酉、正交和辛对称矩阵系综的软边间隙概率，从而恢复了Baik-Deift-Johansson和Baik-Rains的定理.
Random permutations using GEPP
本文研究随机矩阵在部分主元高斯消去中诱导的置换，分类蝴蝶矩阵的置换结构，分析GOE、GUE等模型的非均匀性及渐近行为，并揭示LAPACK实现细节对主元概率的影响，建立随机矩阵与置换组合学的新联系。
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