Integer sequences and k-commuting permutations

Let $\beta$ be any permutation on $n$ symbols and let $c(k,\beta )$ be the number of permutations that $k$-commute with $\beta$. The cycle type of a permutation $\beta$ is a vector $(c_1,\dots ,c_n)$ such that $\beta$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k,\beta )$, for some cycle types. We also express these formulas in terms of integer sequences as given in “The On-line Encyclopedia of Integer Sequences” (OEIS). For some of these sequences we obtain either new interpretations or relationships with sequences in the OEIS database.