Polynomiality of irreducible characters of the symmetric groups

Consider Young diagrams differing only by the length of the first row (i.e., the form of diagrams below the first row is fixed). We prove that the values of the irreducible characters of the groups $\mathrm S_n$ corresponding to these diagrams are given by a polynomial of a special form with respect to natural parameters related to the cycle notation of permutations. Bibl. 3 titles.