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

摘要

We prove that the Stanley-Wilf limit of any layered permutation pattern of length ? is at most 4? ~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. 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.