![]() 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. ![]()
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