Profiles of permutations

TL;DR

This paper explores the cycle structure of restricted and weighted permutations, revealing asymptotic behaviors and probability limits, and draws parallels between different permutation models using probabilistic and combinatorial methods.

Contribution

It establishes a detailed analogy between permutations with restricted cycle lengths and Ewens-distributed permutations, including asymptotic cycle counts and cycle length probabilities.

Findings

Expected number of certain cycles is approximately (1/2) log n

Probability of a cycle longer than a fraction of n approaches a power law

Parallel between Ewens distribution and density-restricted permutations

Abstract

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $\sigma$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all ${1 \over 2} \log n + O(1)$, and the variance is of the same order. Furthermore, we show that in permutations of $[n]$ chosen from the Ewens distribution with parameter $\sigma$, the probability of a random element being in a cycle longer than $\gamma n$ approaches $(1-\gamma)^\sigma$ for large $n$. The same limit law holds for permutations with cycles carrying multiplicative weights with average $\sigma$. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.