Bijective enumeration of some colored permutations given by the product of two long cycles

Let γnγn be the permutation on nn symbols defined by γn=(12…n). We are interested in an enumerative problem on colored permutations, that is permutations ββ of nn in which the numbers from 11 to nn are colored with pp colors such that two elements in a same cycle have the same color. We show that the proportion of colored permutations such that γnβ−1γnβ−1 is a long cycle is given by the very simple ratio 1n−p+1. Our proof is bijective and uses combinatorial objects such as partitioned hypermaps and thorn trees. This formula is actually equivalent to the proportionality of the number of long cycles αα such that γnαγnα has mm cycles and Stirling numbers of size n+1n+1, an unexpected connection previously found by several authors by means of algebraic methods. Moreover, our bijection allows us to refine the latter result with the cycle type of the permutations.