Given a fixed line
L in the
Euclidean plane, a
meander of order
n is a self-avoiding closed curve in the plane that crosses the line at 2
n points. Two meanders are equivalent if one meander can be
continuously deformed into the other while maintaining its property of being a meander and leaving the order of the bridges on the road, in the order in which they are crossed, invariant.
Examples The single meander of order 1 intersects the line twice: : This meander intersects the line four times and thus has order 2: : There are two meanders of order 2. Flipping the image vertically produces the other. There are eight non-equivalent meanders of order 3, each intersecting the line six times. Here are two of them: :
Meandric numbers The number of distinct meanders of order
n is the
meandric number Mn. The first fifteen meandric numbers are given below . :
M1 = 1 :
M2 = 2 :
M3 = 8 :
M4 = 42 :
M5 = 262 :
M6 = 1828 :
M7 = 13820 :
M8 = 110954 :
M9 = 933458 :
M10 = 8152860 :
M11 = 73424650 :
M12 = 678390116 :
M13 = 6405031050 :
M14 = 61606881612 :
M15 = 602188541928
Meandric permutations A
meandric permutation of order
n is defined on the set {1, 2, ..., 2
n} and is determined as follows: • With the line oriented from left to right, each intersection of the meander is consecutively labelled with the integers, starting at 1. • The curve is oriented upward at the intersection labelled 1. • The
cyclic permutation with no fixed points is obtained by following the oriented curve through the labelled intersection points. In the diagram on the right, the order 4 meandric permutation is given by (1 8 5 4 3 6 7 2). This is a
permutation written in
cyclic notation and not to be confused with
one-line notation. If π is a meandric permutation, then π2 consists of two
cycles, one containing all the even symbols and the other all the odd symbols. Permutations with this property are called
alternate permutations, since the symbols in the original permutation alternate between odd and even integers. However, not all alternate permutations are meandric because it may not be possible to draw them without introducing a self-intersection in the curve. For example, the order 3 alternate permutation, (1 4 3 6 5 2), is not meandric. ==Open meander==