1324-avoiding permutations revisited

Abstract

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in $14$ further terms of the generating function, which is now known for all patterns of length $\le 50$. We re-analyse the generating function and find additional evidence for our earlier conclusion that unlike other classical length-$4$ pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of $1324$-avoiding permutations of length $n$ behaves as \[ B\cdot μ^n \cdot μ_1^{\sqrt{n}} \cdot n^g. \] We estimate $\mu =11.600\pm 0.003$, ${\mu }_1=0.0400\pm 0.0005$, $g=-1.1\pm 0.1$ while the estimate of $B$ depends sensitively on the precise value of $\mu$, ${\mu }_1$ and $g$. This reanalysis provides substantially more compelling arguments for the presence of the stretched exponential term ${\mu }_1^{\sqrt{n}}$.