Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

12 pages. This is the final version for FPSAC 2010 proceedingsInternational audienceWe look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1\ 2\ \ldots\ N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account