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Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\2\...\ n)$. We are interested in an enumerative problem on coloredpermutations, that is permutations $\beta$ of $n$ in which the numbers from 1to $n$ are colored with $p$ colors such that two elements in a same cycle havethe same color. We show that the proportion of colored permutations such that$\gamma_n \beta^{-1}$ is a long cycle is given by the very simple ratio$\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects suchas partitioned hypermaps and thorn trees. This formula is actually equivalentto the proportionality of the number of long cycles $\alpha$ such that$\gamma_n\alpha$ has $m$ cycles and Stirling numbers of size $n+1$, anunexpected connection previously found by several authors by means of algebraicmethods. Moreover, our bijection allows us to refine the latter result with thecycle type of the permutations.

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