Explicit expression for the product of the classes of fixed-point free involutions

The structure coefficients of the center of the symmetric group algebra are those appearing when writing the product of conjugacy classes of the symmetric group as a linear combination of conjugacy classes.It is well known that obtaining explicit formulae for particular cases of these coefficients is a difficult problem.In this paper, we recall some motivations for computing the structure coefficients appearing in the product of two equal conjugacy classes.We provide explicit formulas for the productsare the formal sums of the permutations in the conjugacy classes of fixed-point free involutions, involutions with exactly one fixed point and involutions with exactly two fixed points, respectively.