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 $B =9.5 \pm 1.0.$Comment: 20 pages, 10 figure