Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

We prove that the Stanley-Wilf limit of any layered permutation pattern of length ℓ is at most 4ℓ2, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern.We also conjecture that, for any k⩾0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n+1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most eπ2/3≃13.001954.