The Number of $k$-Cycles In a Family of Restricted Permutations

TL;DR

This paper investigates the cycle structure of permutations under specific restrictions, proving a normal distribution for the number of k-cycles and extending results to compositions of n.

Contribution

It introduces new asymptotic normality results for cycle counts in restricted permutations and related compositions, expanding understanding of permutation structures under constraints.

Findings

Number of k-cycles follows a normal distribution asymptotically.

Results extend to CLTs for fixed-size parts in compositions.

Provides new insights into restricted permutation cycle structures.

Abstract

In this paper we study different restrictions imposed over the set of permutations of size $n$, $S_n$, and for specific classes of restrictions study the cycle structure of corresponding permutations. More specifically, we prove that for any fixed positive integer $k$, the number of $k$-cycles of a uniformly chosen permutation $\pi \in S_n$ with the restriction "$\pi(i) \geq i-1$" for $i \in \{2, \ldots, n\}$ has a Normal asymptotic distribution. We further prove that this result translates into CLTs regarding multiplicities of fixed-size parts of a uniformly selected composition of $n$.