1/5/2024 0 Comments To meander definition![]() ![]() The first fifteen semi-meandric numbers are given below (sequence A000682 in the OEIS). The number of distinct semi-meanders of order n is the semi-meandric number M n (usually denoted with an overline instead of an underline). The semi-meander of order 2 intersects the ray twice: The semi-meander of order 1 intersects the ray once: ![]() 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. 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. The first fifteen open meandric numbers are given below (sequence A005316 in the OEIS). The number of distinct open meanders of order n is the open meandric number m n. The open meander of order 2 intersects the line twice: The open meander of order 1 intersects the line once: 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 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. For example, the order 3 alternate permutation, (1 4 3 6 5 2), is not meandric. However, not all alternate permutations are meandric because it may not be possible to draw them without introducing a self-intersection in the curve. Permutations with this property are called alternate permutations, since the symbols in the original permutation alternate between odd and even integers. If π is a meandric permutation, then π 2 consists of two cycles, one containing of all the even symbols and the other all the odd symbols. This is a permutation written in cyclic notation and not to be confused with one-line notation. In the diagram on the right, the order 4 meandric permutation is given by (1 8 5 4 3 6 7 2). The cyclic permutation with no fixed points is obtained by following the oriented curve through the labelled intersection points.The curve is oriented upward at the intersection labelled 1.With the line oriented from left to right, each intersection of the meander is consecutively labelled with the integers, starting at 1.The first fifteen meandric numbers are given below (sequence A005315 in the OEIS). The number of distinct meanders of order n is the meandric number M n. Here are two non-equivalent meanders of order 3, each intersecting the line six times: Flipping the image vertically produces the other. This meander intersects the line four times and thus has order 2: The single meander of order 1 intersects the line twice: 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. 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. The points where the line and the curve cross are therefore referred to as "bridges". Intuitively, a meander can be viewed as a meandering river with a straight road crossing the river over a number of bridges. 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. ![]()
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