  We study asymptotics of characters of the symmetric groups on a fixed
conjugacy class. It was proved by Kerov that such a character can be expressed
as a polynomial in free cumulants of the Young diagram (certain functionals
describing the shape of the Young diagram). We show that for each genus there
exists a universal symmetric polynomial which gives the coefficients of the
part of Kerov character polynomials with the prescribed homogeneous degree. The
existence of such symmetric polynomials was conjectured by Lassalle.
