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 4 ℓ 2 , 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. o 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 e π 2 / 3 ≃ 13.001954 .