On 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 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 n · µ n σ 1 · n g .We estimate µ = 11.60 ± 0.01, σ = 1/2, µ 1 = 0.0398 ± 0.0010, g = −1.1 ± 0.2 and B = 9.5 ± 1.0.1

1324-avoiding permutations revisited

“…We can get somewhat more rapid convergence of the ratios if we remove the term O(1/n) from eqn. (4), and this we do by calculating the modified ratios l n = n · r n − (n − 1) · r n−1 = µ 1 + log…”

“…In an earlier paper [4], two of the current authors gave further coefficients and a detailed analysis of the generating function for 1324 pattern-avoiding permutations (PAPs), extending the known ordinary generating function (OGF) by a further 5 terms. That analysis led us to conjecture that, unlike the known length-4 PAPs, notably the classes Av(1234) [12] and Av(1342) [2], the OGF for Av(1324) included a stretched exponential term.…”

A structural characterisation of Av(1324) and new bounds on its growth rate

“…With the help of computers, | Av n (1324)| has been determined for all n 50. Conway, Guttmann and Zinn-Justin [14,15] have analysed the numbers and give a numerical estimate for gr(Av(1324)) of µ ≈ 11.600 ± 0.003. They also conjecture that | Av n (1324)| behaves asymptotically as A·µ n ·λ √ n ·n α , for certain estimated constants A, λ and α.…”

Staircases, dominoes, and the growth rate of 1324-avoiders

“…Indeed, during a conference in 2004 when the result of [1] was announced, Doron Zeilberger famously declared that "not even God knows S 1000 (1324)". Humans, with the help of computers, now know S 36 (1324), and Conway and Guttman's analysis [13] of their computation provides an estimate for gr(Av(1324)) of 11.60±0.01, and they conjecture that…”