Let P(n,j) denote the number of j-permutations of n objects. In this paper we obtain the generating function for the alternating sequence (-1)^j P(n,j). Our method gives an integral representation for the difference D_n – n!/e, where D_n denotes the number of derangements on n objects. Using this integral representation, we compute the moments of this difference, and we also get an asymptotic expansion for D_n with coefficients in terms of the Bell numbers B_n. We also give a simple proof of the irrationality of e.