Meander (mathematics)

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".

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 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.

Examples

The single meander of order 1 intersects the line twice:

This meander intersects the line four times and thus has order 2:

There are two meanders of order 2. Flipping the image vertically produces the other.

Here are two non-equivalent meanders of order 3, each intersecting the line six times:

Meandric numbers

The number of distinct meanders of order n is the meandric number M_n. The first fifteen meandric numbers are given below (sequence A005315 in the OEIS).

M₁ = 1

M₂ = 2

M₃ = 8

M₄ = 42

M₅ = 262

M₆ = 1828

M₇ = 13820

M₈ = 110954

M₉ = 933458

M₁₀ = 8152860

M₁₁ = 73424650

M₁₂ = 678390116

M₁₃ = 6405031050

M₁₄ = 61606881612

M₁₅ = 602188541928

Meandric permutations

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.

Open meander

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.

Examples

The open meander of order 1 intersects the line once:

The open meander of order 2 intersects the line twice:

Open meandric numbers

The number of distinct open meanders of order n is the open meandric number m_n. The first fifteen open meandric numbers are given below (sequence A005316 in the OEIS).

m₁ = 1

m₂ = 1

m₃ = 2

m₄ = 3

m₅ = 8

m₆ = 14

m₇ = 42

m₈ = 81

m₉ = 262

m₁₀ = 538

m₁₁ = 1828

m₁₂ = 3926

m₁₃ = 13820

m₁₄ = 30694

m₁₅ = 110954

Semi-meander

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.

Examples

The semi-meander of order 1 intersects the ray once:

The semi-meander of order 2 intersects the ray twice:

Semi-meandric numbers

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 first fifteen semi-meandric numbers are given below (sequence A000682 in the OEIS).

M₁ = 1

M₂ = 1

M₃ = 2

M₄ = 4

M₅ = 10

M₆ = 24

M₇ = 66

M₈ = 174

M₉ = 504

M₁₀ = 1406

M₁₁ = 4210

M₁₂ = 12198

M₁₃ = 37378

M₁₄ = 111278

M₁₅ = 346846

Properties of meandric numbers

There is an injective function from meandric to open meandric numbers:

M_n = m_2n−1

Each meandric number can be bounded by semi-meandric numbers:

M_n ≤ M_n ≤ M_2n

For n > 1, meandric numbers are even:

M_n ≡ 0 (mod 2)

External links

"Approaches to the Enumerative Theory of Meanders" by Michael La Croix
P. Di Francesco; O. Golinelli; E. Guitter (October–November 1997). "Meander, folding, and arch statistics". Mathematical and Computer Modelling. 26 (8–10): 97–147. arXiv:hep-th/9506030. doi:10.1016/S0895-7177(97)00202-1.