# 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.

## Key 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.