  Kerov's polynomials give irreducible character values in term of the free
cumulants of the associated Young diagram. We prove in this article a
positivity result on their coefficients, which extends a conjecture of S.
Kerov. Our method, through decomposition of maps, gives a description of the
coefficients of the k-th Kerov's polynomials using permutations in S(k). We
also obtain explicit formulas or combinatorial interpretations for some
coefficients. In particular, we are able to compute the subdominant term for
character values on any fixed permutation (it was known for cycles).
