A transfer matrix approach to the enumeration of plane meanders

Abstract

A closed plane meander of order n is a closed self-avoiding curve intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.

Bounds for the growth rate of meander numbers

“…This interpretation is implicit in a number of earlier works and is similar to the description of meanders by means of "configurations" in [9] and also to the description of meanders in terms of Motzkin words found in [13].…”

“…Two meanders are equivalent if one can be transformed into the other by continuous deformations of the plane, which leave the line fixed (as a set). A number of authors have addressed the problem of exact and asymptotic enumeration of the number M n of meanders of order n (see for instance [6,9] and references therein). The relationship between the enumeration of meanders and Hilbert's 16th problem is discussed in [1] and a general survey of the connection between meanders and related structures and problems in mathematics and physics can be found in [2].…”

Cycles in Random Meander Systems

“…These numbers are confirmed in [11]. In [15] and [16], the calculations have been extended to R 24 by using a different method. Table 1 in [16] gives R 24 = 794337831754564188184.…”

“…n=0 n z n , have real non-negative coefficients y 0 = 0, y n ≥ 0.Definition A.1. An analytic GF y(z) is said to have a stable dominant singularity 3 at z = r > 0, if there exists a bivariate function G(z, w) such that(15) y(z) = G(z, y(z)),…”

A fast algorithm to generate open meandric systems and meanders

“…Meanders are also related to simple alternating transit mazes of depth n [Phillips 1989], and to ovals of planar algebraic curves (Hilbert's 16th problem) [Arnold 1988]. The problem of enumerating meanders is known to be a difficult problem and a significant body of work has been dedicated to it [Albert and Paterson 2005;Francesco et al 1996;Franz and Earnshaw 2002;Jensen 2000;Lando and Zvonkin 1992]. However, until now, no algorithms have been developed that focus on the efficient generation of meanders.…”