Meander (mathematics)

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In mathematics, a meander or closed meander is a self-avoiding closed curve 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.

Contents 1 Meander 1.1 Examples 1.2 Meandric numbers 2 Open meander 2.1 Examples 2.2 Open meandric numbers 3 Semi-meander 3.1 Examples 3.2 Semi-meandric numbers 4 Properties of meandric numbers

[edit] Meander

Given a fixed oriented line L in the Euclidean plane R², a meander of order n is a non-self-intersecting closed curve in R² which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

[edit] Examples

The meander of order 1 intersects the line twice:

The meanders of order 2 intersect the line four times:

[edit] 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 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

[edit] Open meander

Given a fixed oriented line L in the Euclidean plane 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 plane.

[edit] Examples

The open meander of order 1 intersects the line once:

The open meander of order 2 intersects the line twice:

[edit] 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 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

[edit] Semi-meander

Given a fixed oriented ray R in the Euclidean plane 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 homeomorphic in the plane.

[edit] Examples

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

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

[edit] 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 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

[edit] 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)