Small configurations in simple permutations

We prove that every entry in a simple permutation of length at least 4 is contained in a copy of 2413 or 3412 or plays the role of the ‘3’ in a copy of 25 314 or 41 352. This allows us to give a short proof that the number of permutations with at most r copies of 132 has an algebraic generating function for any integer r .