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 coecients and nd 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 n n n g : We estimate = 11:60 0:01; = 1=2; 1 = 0:0398 0:0010; g = 1:1 0:2 and B = 9:5 1:0: