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Central cut is a 1-1 cut at the center of the meander (the i-line is for i=n).

A meander together with the horizontal line separates the plane into several connected components. Each component has a given number of edges which is always an even number. The digons (or bigons) are the faces with least number of edges, that is 2. Equivalently, the number of digons is the number of arches between adjacent sites ("minimal arches") where the two extremal ones are considered adjacent.

A forest meander system is a meander system that does not have any components which are entirely enclosed by another. An equivalent condition is that all components have their least point at an odd index (if points are numbered from 1). The greatest point will then be at an even index.

Exactly half of all meander systems with two components are forest meander systems. This is because when the meander's permutation is rotated one step at a time, one meander will be enclosed in the other on every second step.

a(n) counts closed plane meanders according to the number of white regions when regions are colored black and white alternatively. So the sum of each row is given by A005315. The outer columns consist of 1's. The next-to-outer columns are given by A002415.

This is also the number of arches above the x-axis going from an odd vertex to a higher even vertex(p) for closed plane meanders(M) with n arches. By symmetry, these same subsets exist for arches below the x-axis. For each meander solution, the total arches for the top and bottom that go from an odd vertex to a higher even vertex is n+1.

Example: M(n,p): M(3,1)=1 [(top 16,23,45; bottom 12,34,56)], M(3,2)=6 [(top 14,23,56; bottom 16,25,34)(top 16,25,34; bottom 14,23,56) (top 12,36,45; bottom 16,25,34) (top 16,25,34; bottom 12,36,45) (top 12,36,45; bottom 14,23,56)(top 14,23,56; bottom 12,36,45)] M(3,3)=1 [(top 12,34,56; bottom 16,23,45)]. - Roger Ford, Sep 29 2014

There is no obligation to cross the lower road (cf. A077054).

A tight meander of width n is a special kind of meander defined as follows.

For any pair (S={s_1,...,s_k},T={t_1,...,t_l}) of subsets of {1,...,n-1} (k or l might be 0), the tight meander M(S,T) defined by (S,T) is the following subset of R^2:

assuming S and T ordered so that 0=s_0

semicircles in the upper half-plane with endpoints (s_{i-1}+j,0) and (s_i+1-j,0), for i=1,...,k+1, and j positive integer with s_{i-1}+j

and semicircles in the lower half-plane with endpoints (t_{i-1}+j,0) and (t_i+1-j,0), for i=1,...,l+1, and j positive integer with t_{i-1}+j

The tight meander M(S,T) is called contractible if it is a contractible subspace of R^2, i.e., is either a single point or homeomorphic to an interval.

Then, a(n) is the number of pairs (S,T) as above such that the tight meander M(S,T) is contractible.

From Roger Ford, Jul 05 2023: (Start)

The following is a definition for closed meanders that yield the same sequence as tight meanders. T(n,k) = the number of closed meanders with n top arches and with k exterior arches and k arches of length 1.

e = exterior arch (arch with no covering arch), 1 = arch with length 1, e1 = arch that is exterior with a length of 1:

e exterior length 1

__________ arches arches

/ ____ \

e1 / / \ \ top = 2 top = 2

/\ / / /\1 \ \

/ \ / / / \ \ \

\ \ / / \ \ / / bottom = 2 bottom = 2

\ \/1 / \ \/1 / total = 4 total = 4

\____/ \____/

e e Example T(4,4).

(End)

			For n=3 the a(3)=6 contractible tight meanders of width 3 correspond to the following pairs of subsets of {1,2}: ({},{1}), ({},{2}), ({1},{}), ({2},{}), ({1},{2}), ({2},{1}).