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

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, 310, 30, 1, 0, 13820, 22582, 14196, 4270, 630, 42, 1, 0, 110954, 203006, 149024, 55524, 11060, 1148, 56, 1, 0, 933458, 1886004, 1577712, 698952, 175560, 25032, 1932, 72, 1

Offset: 0

Author

Andrew Howroyd, Jan 31 2025

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins:
  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,  310,  30,  1;
  0, 13820, 22582, 14196, 4270, 630, 42, 1;
  ...
The T(3,2) = 6 forest meander systems are the following and their reflections.
       ______
      / ____ \                 ___
     / /    \ \               /   \
 .. / /. /\ .\ \ ..   and .. / / \ \ . /\ ..
    \/   \/   \/             \/   \/   \/
        (2)                     (4)
.
There are also 6 systems that are not forest meander systems:
      ____                    ______
     / __ \                  /      \
 .. / /  \ \ ..      and .. / /\  /\ \ ..
    \ \/\/ /                \ \/ /  \/
     \____/                  \__/
       (2)                     (4)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows 0..20)
• Roland Bacher, Meander algebras.

Crossrefs

Row sums are A060148.

Column k=1 is A005315.

Column k=2 is half of A006657.

Main diagonal is A000012.

Second diagonal is A002378.

Cf. A008828 (all meander systems), A060174, A060198.