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We prove that the Stanley-Wilf limit of any layered permutation pattern oflength $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of thepattern 1324 is at most 16. These bounds follow from a more general resultshowing that a permutation avoiding a pattern of a special form is a merge oftwo permutations, each of which avoids a smaller pattern. If the conjecture istrue that the maximum Stanley-Wilf limit for patterns of length $\ell$ isattained by a layered pattern then this implies an upper bound of $4\ell^2$ forthe Stanley-Wilf limit of any pattern of length $\ell$.  We also conjecture that, for any $k\ge 0$, the set of 1324-avoidingpermutations with $k$ inversions contains at least as many permutations oflength $n+1$ as those of length $n$. We show that if this is true then theStanley-Wilf limit for 1324 is at most $e^{\pi\sqrt{2/3}} \simeq 13.001954$.

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