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I. Experimental. The calculation of maze numbers has been carried much further by Iwan Jensen and Anthony J. Guttmann of the University of Melbourne, using an algorithm based on transfer matrix methods. Here are the numbers they published in Critical exponents of plane meanders.
Table 1. The number $M_n$ of connected closed meanders with $2n$ crossings.
$$\begin{array}{rl} 1&1 \\ 2& 2 \\ 3& 8 \\ 4& 42\\ 5& 262\\ 6& 1 828 \\ 7& 13 820 \\ 8& 110 954 \\ 9& 933 458 \\ 10 & 8 152 860 \\ 11& 73 424 650 \\ 12& 678 390 116 \\ 13& 6 405 031 050 \\ 14 & 61 606 881 612 \\ 15 & 602 188 541 928 \\ 16 & 5 969 806 669 034 \\ 17& 59 923 200 729 046 \\ 18& 608 188 709 574 124 \\ 19& 6 234 277 838 531 806 \\ 20& 64 477 712 119 584 604 \\ 21& 672 265 814 872 772 972 \\ 22& 7 060 941 974 458 061 392 \\ 23& 74 661 728 661 167 809 752 \\ 24& 794 337 831 754 564 188 184 \end{array}$$ "The number of closed meanders is expected to grow exponentially, with a sub-dominant term given by a critical exponent, $$M_n \sim C R^{2n}/n^{\alpha}$$ The exponential growth constant $R$ is often called the connective constant", while $\alpha$ is the "coefficient exponent."
Using these and other data, Jensen and Guttmann estimate the constants as $R=3.501837(3)$ and $\alpha=3.4208(6)$
II. Theoretical. P. Di Francesco, O. Golinelli and E. Guitter, of the Service de Physique Théorique at Saclay have a sequence of papers culminating in Meanders: exact asymptotics. There they propose a model from conformal field theory ("the gravitational version of a c=-4 two-dimensional conformal field theory") which allows them to conjecture an exact limit for the Meander coefficient exponent. Their number $\sqrt{29}(\sqrt{29}+\sqrt{5})/12 = 3.4201328\dots $ is in agreement with the Australian team's estimates.
M.H. Albert (Computer Science, University of Otago) and M.S. Patterson
(Computer Science, University of Warwick) have published
Bounds
for the Growth Rate of Meander Numbers (2004). They define
$M_n$ to be the number of meanders which cross the vertical axis at
$2n$ points, and seek upper and lower bounds on the exponential part
of the asymptotic form.
They observe that $M=\lim_{n\rightarrow\infty}M_n^{1/n}$
exists and show that
$11.380 < M< 12.901$.
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