Colored permutations with no monochromatic cycles

An (n₁, n₂,…, n_k)-colored permutation is a permutation of n₁ + n₂ + · · · + n_k in which 1, 2,…, n₁ have color 1, and n₁ + 1, n₁ + 2,…, n₁ + n₂ have color 2, and so on. We give a bijective proof of Stein-hardt’s result: the number of colored permutations with no monochro-matic cycles is equal to the number of permutations with no fixed points after reordering the first n₁ elements, the next n₂ element, and so on, in ascending order. We then find the generating function for colored per-mutations with no monochromatic cycles. As an application we give a new proof of the well known generating function for colored permutations with no fixed colors, also known as multi-derangements.