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arXiv · 1709.01248

1324-avoiding permutations revisited

Andrew R. Conway·Anthony J. Guttmann·Paul Zinn-Justin
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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 $μ=11.600 \pm 0.003$, $μ_1 = 0.0400 \pm 0.0005$, $g = -1.1 \pm 0.1$ while the estimate of $B$ depends sensitively on the precise value of $μ$, $μ_1$ and $g$. This reanalysis provides substantially more compelling arguments for the presence of the stretched exponential term $μ_1^{\sqrt{n}}$.

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Andrew R. Conway, Anthony J. Guttmann, Paul Zinn-Justin. 2017-11-18. 1324-avoiding permutations revisited. https://arxiv.org/abs/1709.01248

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