Related papers: Random permutations without macroscopic cycles

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.

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1…

We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild…

We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after suitable…

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the…

We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a…

In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this…

We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities…

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…

We study the asymptotic behavior of the long cycles of a random permutation of $n$ objects with respect to multiplicative measures with polynomial growing cycle weights. We show that the longest cycle and the length differences between the…

It is known that the number of permutations in the symmetric group $S_{2n}$ with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent…

A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from…

We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem,…

Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. For $A(n)=A\cap [n]$, the limit $\rho=\lim_{n\to\infty}\mid A(n)\mid/n$ (if it esists) is…

Let $S_n$ be the set of all permutations of $\{1,2,\ldots,n\}$ and let $\sigma=(\sigma_1,\sigma_2,\ldots,\sigma_n)\in S_n$. The {\it initial longest increasing sequence} (ILIS) in $\sigma$ has length $m$ if, for $1\le m\le n-1$,…

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 study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi \in \mathbb{S}_n$ is proportional to $q^{\textrm{inv}(\pi)}$ where $0<q\le 1$ and…

We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e.,…

A universal cycle for permutations of length $n$ is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length $n$, and containing all permutations of length $n$ as factors. It is well known…

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.