COLORED PERMUTATIONS WITH NO MONOCHROMATIC CYCLES

Abstract

. An ( n 1 ,n 2 ,...,n k )-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 Stein- hardt’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 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.