Математикалық меандрлар және олардың түрлері

In mathematics, a meander or closed meander is a self avoiding closed curve which crosses a given line a number of times, meaning that it intersects the line while passing from one side to the other. Intuitively, a meander can be viewed as a meandering river with a straight road crossing the river over a number of bridges. The points where the line and the curve cross are therefore referred to as "bridges".

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 2n 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.

A meandric permutation of order n is defined on the set {1, 2, , 2n} 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 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 line L in the Euclidean plane, an open meander of order n is a non self intersecting curve in the plane that crosses the line at n points. Two open meanders are equivalent if one can be continuously deformed into the other while maintaining its property of being an open meander and leaving the order of the bridges on the road, in the order in which they are crossed, invariant.

Given a fixed oriented ray R (a closed half line) in the Euclidean plane, a semi meander of order n is a non self intersecting closed curve in the plane that crosses the ray at n points. Two semi meanders are equivalent if one can be continuously deformed into the other while maintaining its property of being a semi meander and leaving the order of the bridges on the ray, in the order in which they are crossed, invariant.