On the growth rate of 1324-avoiding permutations

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 patternavoiding 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 · μ · μ σ 1 · n. We estimate μ = 11.60 ± 0.01, σ = 1/2, μ1 = 0.0398 ± 0.0010, g = −1.1 ± 0.2 and B = 9.5± 1.0.