 School of Mathematics and Statistics  Research Publications
School of Mathematics and Statistics  Research Publications
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ItemClassical Length5 PatternAvoiding PermutationsClisby, N ; Conway, AR ; Guttmann, AJ ; Inoue, Y (The Electronic Journal of Combinatorics, 20220101)We have made a systematic numerical study of the 16 Wilf classes of length5 classical patternavoiding permutations from their generating function coefficients. We have extended the number of known coefficients in fourteen of the sixteen classes. Careful analysis, including sequence extension, has allowed us to estimate the growth constant of all classes, and in some cases to estimate the subdominant powerlaw term associated with the exponential growth. In six of the sixteen classes we find the familiar powerlaw behaviour, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot n^g,$ while in the remaining ten cases we find a stretched exponential as the most likely subdominant term, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where $0 < \sigma < 1.$ We have also classified the 120 possible permutations into the 16 distinct classes. We give compelling numerical evidence, and in one case a proof, that all 16 Wilfclass generating function coefficients can be represented as moments of a nonnegative measure on $[0,\infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as logconvexity, which can be used to provide quite strong rigorous lower bounds. Stronger bounds still can be established under plausible monotonicity assumptions about the terms in the continuedfraction expansion of the generating functions implied by the Stieltjes property. In this way we provide strong (nonrigorous) lower bounds to the growth constants, which are sometimes within a few percent of the exact value.

ItemAsymptotics of 3StackSortable PermutationsDefant, C ; Price, AE ; Guttmann, AJ (ELECTRONIC JOURNAL OF COMBINATORICS, 20210618)We derive a simple functional equation with two catalytic variables characterising the generating function of 3stacksortable permutations. Using this functional equation, we extend the 174term series to 1000 terms. From this series, we conjecture that the generating function behaves as $$W(t) \sim C_0(1\mu_3 t)^\alpha \cdot \log^\beta(1\mu_3 t), $$ so that $$[t^n]W(t)=w_n \sim \frac{c_0\mu_3^n}{ n^{(\alpha+1)}\cdot \log^\lambda{n}} ,$$ where $\mu_3 = 9.69963634535(30),$ $\alpha = 2.0 \pm 0.25.$ If $\alpha = 2$ exactly, then $\lambda = \beta+1$, and we estimate $\beta \approx 2,$ but with a wide uncertainty of $\pm 1.$ If $\alpha$ is not an integer, then $\lambda=\beta$, but we cannot give a useful estimate of $\beta$. The growth constant estimate (just) contradicts a conjecture of the first author that $$9.702 < \mu_3 \le 9.704.$$ We also prove a new rigorous lower bound of $\mu_3\geq 9.4854$, allowing us to disprove a conjecture of Bóna. We then further extend the series using differentialapproximants to obtain approximate coefficients $O(t^{2000}),$ expected to be accurate to $20$ significant digits, and use the approximate coefficients to provide additional evidence supporting the results obtained from the exact coefficients.

ItemSelfavoiding walks in a rectangleGuttmann, AJ ; Kennedy, T (SPRINGER, 20140201)

ItemThe Critical Fugacity for Surface Adsorption of SelfAvoiding Walks on the Honeycomb Lattice isBeaton, NR ; BousquetMelou, M ; de Gier, J ; DuminilCopin, H ; Guttmann, AJ (SPRINGER, 20140301)

ItemCompressed selfavoiding walks, bridges and polygonsBeaton, NR ; Guttmann, AJ ; Jensen, I ; Lawler, GF (IOP PUBLISHING LTD, 20151113)

ItemFuchsian differential equation for the perimeter generating function of threechoice polygonsGUTTMANN, A ; JENSEN, I ( 2005)

ItemScaling function and universal amplitude combinations for selfavoiding polygonsRichard, C ; Guttmann, AJ ; Jensen, I (IOP PUBLISHING LTD, 20010914)

ItemThe perimeter generating function of punctured staircase polygonsGuttmann, AJ ; Jensen, I (IOP PUBLISHING LTD, 20060414)

ItemScaling prediction for selfavoiding polygons revisitedRichard, C ; Jensen, I ; Guttmann, AJ (IOP PUBLISHING LTD, 20040801)

ItemVicious walkers, friendly walkers, and young tableaux. III. Between two wallsKrattenthaler, C ; Guttmann, AJ ; Viennot, XG (SPRINGER, 20030301)