Editing Meander (mathematics) (section)

==Meander==

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 [[Continuous mapping|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:
:[[File:meander_example_1_2.svg|64px]]

This meander intersects the line four times and thus has order 2:
:[[File:meander_example_1_2_3_4.svg|64px]]
There are two meanders of order 2. Flipping the image vertically produces the other.

There are three non-equivalent meanders of order 3, each intersecting the line six times. Here are two of them:
:[[File:meander_example_1_2_3_4_5_6.svg|64px]] [[File:Meander example 1 6 3 4 5 2.svg|64px]]

===Meandric numbers===

The number of distinct meanders of order ''n'' is the '''meandric number''' ''M<sub>n</sub>''. The first fifteen meandric numbers are given below {{OEIS|id=A005315}}.

:''M''<sub>1</sub> = 1
:''M''<sub>2</sub> = 2 
:''M''<sub>3</sub> = 8
:''M''<sub>4</sub> = 42
:''M''<sub>5</sub> = 262
:''M''<sub>6</sub> = 1828
:''M''<sub>7</sub> = 13820
:''M''<sub>8</sub> = 110954
:''M''<sub>9</sub> = 933458
:''M''<sub>10</sub> = 8152860
:''M''<sub>11</sub> = 73424650
:''M''<sub>12</sub> = 678390116
:''M''<sub>13</sub> = 6405031050
:''M''<sub>14</sub> = 61606881612
:''M''<sub>15</sub> = 602188541928

===Meandric permutations===
[[File:meander_example_1_8_5_4_3_6_7_2.svg|thumb|{{center|Meandric permutation <br />(1 8 5 4 3 6 7 2)}}]]
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 π<sup>2</sup> consists of two [[cyclic permutation|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.