  For a finite group $G,$ we define the concept of $G$-partial permutation and
use it to show that the structure coefficients of the center of the wreath
product $G\wr \mathcal{S}_n$ algebra are polynomials in $n$ with non-negative
integer coefficients. Our main tool is a combinatorial algebra which projects
onto the center of the group $G\wr \mathcal{S}_n$ algebra for every $n.$ This
generalizes the Ivanov and Kerov method to prove the polynomiality property for
the structure coefficients of the center of the symmetric group algebra.
