A006657 -id:A006657 - cp's OEIS frontend

A008828 Triangle read by rows: T(n,k) = number of closed meander systems of order n with k<=n components.

1, 2, 2, 8, 12, 5, 42, 84, 56, 14, 262, 640, 580, 240, 42, 1828, 5236, 5894, 3344, 990, 132, 13820, 45164, 60312, 42840, 17472, 4004, 429, 110954, 406012, 624240, 529104, 271240, 85904, 16016, 1430, 933458, 3772008, 6540510, 6413784, 3935238, 1569984, 405552, 63648, 4862

Offset: 1

D. Ivanov, S. K. Lando, A. K. Zvonkin ( LabRI, Bordeaux, France)

A meander of order n has 2n bridges. For many more references, see A005315 and A005316.

			Triangle starts:
   1;
   2  2;
   8 12  5;
  42 84 56 14;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210
Bertrand Duplantier and Emmanuel Guitter, Liouville Quantum Duality and Random Planar Maps, arXiv:2507.12203 [math-ph], 2025. See p. 50.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
Michael La Croix, Approaches to the Enumerative Theory of Meanders [_Gerald McGarvey_, Oct 26 2008]
Sergei K. Lando and Alexander K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
Sergei K. Lando and Alexander K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117 (1993) p. 232.

Columns include A005315, A006657, A006658. Diagonals include A000108 (Catalan numbers), A006659, A007746. Row sums are in A001246.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 10 2004
Edited by Ralf Stephan, Dec 29 2004
T(10,k)-T(20,k) from Andrew Howroyd, Nov 22 2015

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

Original entry on oeis.org

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

Andrew Howroyd, Jan 31 2025

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.

			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)

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

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.

A006658 Closed meanders with 3 components and 2n bridges.

Original entry on oeis.org

5, 56, 580, 5894, 60312, 624240, 6540510, 69323910, 742518832, 8028001566, 87526544560, 961412790002, 10630964761766, 118257400015312, 1322564193698320, 14863191405246888, 167771227744292160, 1901345329566422790

Offset: 3

N. J. A. Sloane

nonn

S. K. Lando and A. K. Zvonkin "Plane and projective meanders", Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)

Cf. A005315, A006657.

A column of triangle A008828.

A008828 = Import["https://oeis.org/A008828/b008828.txt", "Table"][[All, 2]];
a[n_] := A008828[[(n^2 - n + 6)/2]];
a /@ Range[3, 20] (* Jean-François Alcover, Sep 25 2019 *)

a(13)-a(20) from Andrew Howroyd, Nov 22 2015

cp's OEIS Frontend

A008828 Triangle read by rows: T(n,k) = number of closed meander systems of order n with k<=n components.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A006658 Closed meanders with 3 components and 2n bridges.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Extensions