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A380368 Triangle read by rows: T(n,k) is the number of closed forest meander systems with 2n crossings and k components.

%I A380368 #22 Feb 03 2025 14:04:19
%S A380368 1,0,1,0,2,1,0,8,6,1,0,42,42,12,1,0,262,320,130,20,1,0,1828,2618,1360,
%T A380368 310,30,1,0,13820,22582,14196,4270,630,42,1,0,110954,203006,149024,
%U A380368 55524,11060,1148,56,1,0,933458,1886004,1577712,698952,175560,25032,1932,72,1
%N A380368 Triangle read by rows: T(n,k) is the number of closed forest meander systems with 2n crossings and k components.
%C A380368 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.
%C A380368 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.
%H A380368 Andrew Howroyd, <a href="/A380368/b380368.txt">Table of n, a(n) for n = 0..230</a> (rows 0..20)
%H A380368 Roland Bacher, <a href="http://www-fourier.ujf-grenoble.fr/sites/default/files/ref_478.pdf">Meander algebras</a>.
%e A380368 Triangle begins:
%e A380368   1;
%e A380368   0,     1;
%e A380368   0,     2,     1;
%e A380368   0,     8,     6,     1;
%e A380368   0,    42,    42,    12,    1;
%e A380368   0,   262,   320,   130,   20,   1;
%e A380368   0,  1828,  2618,  1360,  310,  30,  1;
%e A380368   0, 13820, 22582, 14196, 4270, 630, 42, 1;
%e A380368   ...
%e A380368 The T(3,2) = 6 forest meander systems are the following and their reflections.
%e A380368        ______
%e A380368       / ____ \                 ___
%e A380368      / /    \ \               /   \
%e A380368  .. / /. /\ .\ \ ..   and .. / / \ \ . /\ ..
%e A380368     \/   \/   \/             \/   \/   \/
%e A380368         (2)                     (4)
%e A380368 .
%e A380368 There are also 6 systems that are not forest meander systems:
%e A380368       ____                    ______
%e A380368      / __ \                  /      \
%e A380368  .. / /  \ \ ..      and .. / /\  /\ \ ..
%e A380368     \ \/\/ /                \ \/ /  \/
%e A380368      \____/                  \__/
%e A380368        (2)                     (4)
%Y A380368 Row sums are A060148.
%Y A380368 Column k=1 is A005315.
%Y A380368 Column k=2 is half of A006657.
%Y A380368 Main diagonal is A000012.
%Y A380368 Second diagonal is A002378.
%Y A380368 Cf. A008828 (all meander systems), A060174, A060198.
%K A380368 nonn,tabl
%O A380368 0,5
%A A380368 _Andrew Howroyd_, Jan 31 2025