Defect of characters of the symmetric group

Following the work of B. Külshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the -defect of characters of the symmetric group n, where is an arbitrary integer. We prove that the -defect is given by an analogue of the hook-length formula, and use it to prove, when, an -version of the McKay conjecture in n.