Summary: A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coefficients of this series are not $P$-recursive, an asymptotic expansion for these coefficients, and a number of congruence results for the coefficients of the functional inverse of the ordinary generating function for all permutations.

05A05, 05A15, 05A16, 11A07

permutation, P -recursiveness, asymptotic enumeration