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

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 "$\pi(i) \geq i-1$\mathrm{"}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 $n$.