Counting stabilized-interval-free permutations

A stabilized-interval-free (SIF) permutation on [n]={1,2,...,n} is one that does not stabilize any proper subinterval of [n]. By presenting a decomposition of an arbitrary permutation into a list of SIF permutations, we show that the generating function A(x) for SIF permutations satisfies the defining property: [x^(n-1)] A(x)^n = n! . We also give an efficient recurrence for counting SIF permutations.