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 **R**^{2}, a **meander** of order *n* is a non-self-intersecting closed curve in **R**^{2} which transversally intersects the line at 2*n* points for some positive integer *n*. Two meanders are said to be equivalent if they are homeomorphic in the plane.
### 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 (sequence A005315 in OEIS). *M*_{1} = 1 *M*_{2} = 2 *M*_{3} = 8 *M*_{4} = 42 *M*_{5} = 262 *M*_{6} = 1828 *M*_{7} = 13820 *M*_{8} = 110954 *M*_{9} = 933458 *M*_{10} = 8152860 *M*_{11} = 73424650 *M*_{12} = 678390116 *M*_{13} = 6405031050 *M*_{14} = 61606881612 *M*_{15} = 602188541928 ## Open meander Given a fixed oriented line *L* in the Euclidean plane **R**^{2}, an **open meander** of order *n* is a non-self-intersecting oriented curve in **R**^{2} 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** *m*_{n}. The first fifteen open meandric numbers are given below (sequence A005316 in OEIS). *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 ray *R* in the Euclidean plane **R**^{2}, a **semi-meander** of order *n* is a non-self-intersecting closed curve in **R**^{2} 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** __M___{n} (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below (sequence A000682 in OEIS). *M*_{1} = 1 *M*_{2} = 1 *M*_{3} = 2 *M*_{4} = 4 *M*_{5} = 10 *M*_{6} = 24 *M*_{7} = 66 *M*_{8} = 174 *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 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) |