k-partial permutations and the center of the wreath product S_k≀ S_n algebra

We generalize the concept of partial permutations of Ivanov and Kerov and introduce k-partial permutations. This allows us to show that the structure coefficients of the center of the wreath product S_k≀ S_n algebra are polynomials in n with nonnegative integer coefficients. We use a universal algebra I∞k, which projects on the center Z(C[S_k≀ S_n]) for each n. We show that I∞k is isomorphic to the algebra of shifted symmetric functions on many alphabets.