mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a meander or closed meander is a self-avoiding

closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges.

Meander

Given a fixed oriented line ''L'' in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

R², a meander of order ''n'' is a non-self-intersecting closed curve in R² which transversally intersects the line at 2''n'' points for some positive integer ''n''. The line and curve together form a meandric system. Two meanders are said to be equivalent if there is a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...

of the whole plane that takes ''L'' to itself and takes one meander to the other.

Examples

The meander of order 1 intersects the line twice: : The meanders of order 2 intersect the line four 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 . :''M''₁ = 1 :''M''₂ = 1 :''M''₃ = 2 :''M''₄ = 8 :''M''₅ = 42 :''M''₆ = 262 :''M''₇ = 1828 :''M''₈ = 13820 :''M''₉ = 110954 :''M''₁₀ = 933458 :''M''₁₁ = 8152860 :''M''₁₂ = 73424650 :''M''₁₃ = 678390116 :''M''₁₄ = 6405031050 :''M''₁₅ = 61606881612

Meandric permutations

A meandric permutation of order ''n'' is defined on the set and is determined by a meandric system in the following way: * 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 In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...

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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

written in

cyclic notation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

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 oriented line ''L'' in the

R², an open meander of order ''n'' is a non-self-intersecting oriented curve in R² which transversally intersects the line at ''n'' points for some positive integer ''n''. Two open meanders are said to be equivalent if they are

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

in the plane.

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 . :''m''₁ = 1 :''m''₂ = 1 :''m''₃ = 2 :''m''₄ = 3 :''m''₅ = 8 :''m''₆ = 14 :''m''₇ = 42 :''m''₈ = 81 :''m''₉ = 262 :''m''₁₀ = 538 :''m''₁₁ = 1828 :''m''₁₂ = 3926 :''m''₁₃ = 13820 :''m''₁₄ = 30694 :''m''₁₅ = 110954

Semi-meander

Given a fixed oriented

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''R'' in the

R², a semi-meander of order ''n'' is a non-self-intersecting closed curve in R² which transversally intersects the ray at ''n'' points for some positive integer ''n''. Two semi-meanders are said to be equivalent if they are

in the plane.

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 . :''M''₁ = 1 :''M''₂ = 1 :''M''₃ = 2 :''M''₄ = 4 :''M''₅ = 10 :''M''₆ = 24 :''M''₇ = 66 :''M''₈ =

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:''M''₉ = 504 :''M''₁₀ = 1406 :''M''₁₁ = 4210 :''M''₁₂ = 12198 :''M''₁₃ = 37378 :''M''₁₄ = 111278 :''M''₁₅ = 346846

Properties of meandric numbers

There is an

injective function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

from meandric to open meandric numbers: :''M_n'' = ''m''_2''n''−1 Each meandric number can be bounded by semi-meandric numbers: :''M_n'' ≤ ''M_n'' ≤ ''M''_2''n'' For ''n'' > 1, meandric numbers are

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...

: :''M_n'' ≡ 0 (mod 2)

External links

"Approaches to the Enumerative Theory of Meanders" by Michael La Croix
{{DEFAULTSORT:Meander (Mathematics) Combinatorics Integer sequences