School of Mathematics and Statistics - Research Publications
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ItemAsymptotics of 3-Stack-Sortable Permutations(ELECTRONIC JOURNAL OF COMBINATORICS, 2021-06-18)We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term 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 differential-approximants 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.
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ItemSelf-avoiding walks in a rectangle(SPRINGER, 2014-02-01)
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ItemCompressed self-avoiding walks, bridges and polygons(IOP PUBLISHING LTD, 2015-11-13)
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ItemScaling function and universal amplitude combinations for self-avoiding polygons(IOP PUBLISHING LTD, 2001-09-14)
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ItemThe perimeter generating function of punctured staircase polygons(IOP PUBLISHING LTD, 2006-04-14)
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ItemScaling prediction for self-avoiding polygons revisited(IOP PUBLISHING LTD, 2004-08-01)
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ItemVicious walkers, friendly walkers, and young tableaux. III. Between two walls(SPRINGER, 2003-03-01)
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ItemCritical behavior of the two-dimensional Ising susceptibility(AMERICAN PHYSICAL SOC, 2001-04-30)We report computations of the short- and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable tau, linear in T/Tc-1, the short-distance terms have the form tau(p)(ln/tau/)q with p> or =q2. A high- and low-temperature series of N = 323 terms, generated using an algorithm of complexity O(N6), are analyzed to obtain the scaling part, which when divided by the leading /tau/(-7/4) singularity contains only integer powers of tau. Contributions of distinct irrelevant variables are identified and quantified at leading orders /tau/(9/4) and /tau/(17/4).