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Integer sequences and k-commuting permutations

Let $β$ be any permutation on $n$ symbols and let $c(k, β)$ be the number of permutations that $k$-commute with $β$. The cycle type of a permutation $β$ is a vector $(c_1, \dots, c_n)$ such that $β$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k, β)$, 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.