On 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 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 asB⋅μn⋅μ1nσ⋅ng. We estimate μ=11.60±0.01, σ=1/2, μ1=0.040±0.0015, g=−1.1±0.2 and B=7±1.3.