Colored permutations with no monochromatic cycles

Abstract

An (n(1), n(2),..., nk)-colored permutation is a permutation of n(1) + n(2) +...+ n(k) in which 1, 2,..., n(1) have color 1, and n(1) + 1, n(1) + 2,..., n(1) + n(2) have color 2, and so on. We give a bijective proof of Steinhardt's result: the number of colored permutations with no monochromatic cycles is equal to the number of permutations with no fixed points after reordering the first n(1) elements, the next n(2) element, and so on, in ascending order. We then find the generating function for colored permutations 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.