| 词条 | Meander (mathematics) |
| 释义 |
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. MeanderGiven 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. The line and curve together form a meandric system. 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. ExamplesThe meander of order 1 intersects the line twice: The meanders of order 2 intersect the line four times. Meandric numbersThe number of distinct meanders of order n is the meandric number Mn. The first fifteen meandric numbers are given below {{OEIS|id=A005315}}. 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 permutationsA meandric permutation of order n is defined on the set {1, 2, ..., 2n} and is determined by a meandric system in the following way:
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 meanderGiven 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. ExamplesThe open meander of order 1 intersects the line once: The open meander of order 2 intersects the line twice: Open meandric numbersThe number of distinct open meanders of order n is the open meandric number mn. The first fifteen open meandric numbers are given below {{OEIS|id=A005316}}. 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-meanderGiven 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. ExamplesThe semi-meander of order 1 intersects the ray once: The semi-meander of order 2 intersects the ray twice: Semi-meandric numbersThe 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 {{OEIS|id=A000682}}. 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 numbersThere is an injective function from meandric to open meandric numbers: Mn = m2n−1 Each meandric number can be bounded by semi-meandric numbers: Mn ≤ Mn ≤ M2n For n > 1, meandric numbers are even: Mn ≡ 0 (mod 2) External links
2 : Combinatorics|Integer sequences |
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