Open meandric number

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.

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 there is a homeomorphism 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 Mn. The first fifteen meandric numbers are given below (sequence 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

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.

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 mn. The first fifteen open meandric numbers are given below (sequence 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 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.

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 Mn (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below (sequence 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:

Mn = m2n−1

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

MnMnM2n

For n > 1, meandric numbers are even:

Mn ≡ 0 (mod 2)

External links

  • "Approaches to the Enumerative Theory of Meanders" by Michael La Croix
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