The Stanely-F\'eray-\'Sniady formula for the generalized characters of the Symmetric Group

We show that the explicit formula of Stanley-F'eray-'Sniady for the characters of the symmetric group have a natural extension to the generalized characters. These are the spherical functions of the unbalanced Gel’fand pair $(S_n\times S_{n-1},\text{diag}{S}_{n-1})$.