Related papers: Profiles of permutations

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}},…

Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and…

Introduced by Mallows as a ranking model in statistics, Mallows permutation model is a class of non-uniform probability distributions on the symmetric group $S_n$. The model depends on a distance metric on $S_n$ and a scale parameter…

We find a two term asymptotic expansion for the optimal expected value of a sequentially selected monotone subsequence from a random permutation of length n. A striking feature of this expansion is that tells us that the expected value of…

Suppose we are given an infinite sequence of input cells, each initialized with a uniform random symbol from $[n]$. How hard is it to output a sequence in $[n]^n$ that is close to a uniform random permutation? Viola (SICOMP 2020)…

There is a growing interest in studying the distribution of certain labels in products of permutations since the work of Stanley addressing a conjecture of B\'{o}na. This paper is concerned with a problem in that direction. Let $D$ be a…

We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive…

We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such…

Normal approximations for descents and inversions of permutations of the set $\{1,2,...,n\}$ are well known. A number of sequences that occur in practice, such as the human genome and other genomes, contain many repeated elements. Motivated…

We study how the order of N independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number m, defined as the number of pairs that are out of sort with respect to the initial…

We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…

Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic…

It is a classical result that a random permutation of $n$ elements has, on average, about $\log n$ cycles. We generalise this fact to all directed $d$-regular graphs on $n$ vertices by showing that, on average, a random cycle-factor of such…

We consider the problem of enumerating permutations in the symmetric group on $n$ elements which avoid a given set of consecutive pattern $S$, and in particular computing asymptotics as $n$ tends to infinity. We develop a general method…

We consider a class of random permutations of the interval $[-n,n]$, in which points are typically displaced a distance $O(W)$. We show the cycles are localized on the scale $W^3$, with an exponentially decaying tail bound. Analogous to…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns in…

We consider nearest neighbour spatial random permutations on $\mathbb{Z}^d$. In this case, the energy of the system is proportional the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually…

We show that almost all permutations have some power that is a cycle of prime length. The proof includes a theorem giving a strong upper bound on the proportion of elements of the symmetric group having no cycles with length in a given set.

In this paper, we study permutations $\pi \in S_n$ with exactly $m$ transpositions. In particular, we are interested in the expected value of $\pi(1)$ when such permutations are chosen uniformly at random. When $n$ is even, this expected…

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough…