  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 $\mathcal{S}_k\wr
\mathcal{S}_n$ algebra are polynomials in $n$ with non-negative integer
coefficients. We use a universal algebra $\mathcal{I}_\infty^k$ which projects
on the center $Z(\mathbb{C}[\mathcal{S}_k\wr \mathcal{S}_n])$ for each $n.$ We
show that $\mathcal{I}_\infty^k$ is isomorphic to the algebra of shifted
symmetric functions on many alphabets.
