On the growth rate of 1324-avoiding permutations

Abstract

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot {\mu }^n\cdot {\mu }_1^{n^{\sigma }}\cdot n^g.$$ We estimate $\mu =11.60\pm 0.01,$ $\sigma =1/2,$ ${\mu }_1=0.0398\pm 0.0010,$ $g=-1.1\pm 0.2$ and $B=9.5\pm \mathrm{1.0.}$

20 pages, 10 figures

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