  Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\
2\...\ n)$. We are interested in an enumerative problem on colored
permutations, that is permutations $\beta$ of $n$ in which the numbers from 1
to $n$ are colored with $p$ colors such that two elements in a same cycle have
the 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 such
as partitioned hypermaps and thorn trees. This formula is actually equivalent
to the proportionality of the number of long cycles $\alpha$ such that
$\gamma_n\alpha$ has $m$ cycles and Stirling numbers of size $n+1$, 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.
