Meandric number
In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges.
Given a fixed oriented line L in the Euclidean plane R2, a meander of order n is a non-self-intersecting closed curve in R2 which transversally intersects the line at 2n points for some positive integer n. The line and curve together form a meandric system. Two meanders are said to be equivalent if there is a homeomorphism of the whole plane that takes L to itself and takes one meander to the other.
The meander of order 1 intersects the line twice:
The meanders of order 2 intersect the line four times:
The number of distinct meanders of order n is the meandric number Mn. The first fifteen meandric numbers are given below (sequence in the OEIS).
A meandric permutation of order n is defined on the set {1, 2, ..., 2n} and is determined by a meandric system in the following way:
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 of 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.
Given a fixed oriented line L in the Euclidean plane R2, an open meander of order n is a non-self-intersecting oriented curve in R2 which transversally intersects the line at n points for some positive integer n. Two open meanders are said to be equivalent if they are homeomorphic in the plane.
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Wikipedia