A077460 -id:A077460 - cp's OEIS frontend

A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.

1, 1, 2, 8, 42, 262, 1828, 13820, 110954, 933458, 8152860, 73424650, 678390116, 6405031050, 61606881612, 602188541928, 5969806669034, 59923200729046, 608188709574124, 6234277838531806, 64477712119584604, 672265814872772972, 7060941974458061392

Offset: 0

N. J. A. Sloane, J. A. Reeds (reeds(AT)idaccr.org)

There is a 1-to-1 correspondence between loops crossing a road 2n times and lines crossing a road 2n-1 times.

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.
S. K. Lando and A. K. Zvonkin, Meanders, Selecta Mathematica Sovietica, Vol. 11, Number 2, pp. 117-144, 1992.
A. Phillips, Simple Alternating Transit Mazes, preprint. Abridged version appeared as "La topologia dei labirinti," in M. Emmer, editor, L'Occhio di Horus: Itinerari nell'Imaginario Matematico. Istituto della Enciclopedia Italia, Rome, 1989, pp. 57-67.
V. R. Pratt, personal communication.
J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For additional references, see A005316.

Andrew Howroyd, Table of n, a(n) for n = 0..28 (first 24 terms from Iwan Jensen)
Oswin Aichholzer, Carlos Alegría Galicia, Irene Parada, Alexander Pilz, Javier Tejel, Csaba D. Tóth, Jorge Urrutia, and Birgit Vogtenhuber, Hamiltonian meander paths and cycles on bichromatic point sets, XVIII Spanish Meeting on Computational Geometry (Girona, 2019).
V. I. Arnol'd, A branched covering of CP^2->S^4, hyperbolicity and projective topology [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.
Roland Bacher, Meander algebras
David Bevan, Random Closed Meanders - _David Bevan_, Jun 25 2010
Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 525.
Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.
Erich Friedman, Illustration of initial terms
Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
Iwan Jensen, Home page
Iwan Jensen, Closed meanders, a(n) for n = 1..24
Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
Iwan Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
Iwan Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).
Michael La Croix, Approaches to the Enumerative Theory of Meanders, 2003.
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)
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018).
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Phillips, Mazes
A. Phillips, Simple, Alternating, Transit Mazes
J. A. Reeds, D. E. Knuth, & N. J. A. Sloane, Email Correspondence
J. Reeds, L. Shepp, & D. McIlroy, Numerical bounds for the Arnol'd "meander" constant, Preprint.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

These are the odd-numbered terms of A005316. Cf. A077054. For nonisomorphic solutions, see A077460.

A column of triangle A008828.

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 1, A005316[[2n]]];
a /@ Range[0, 28] (* Jean-François Alcover, Sep 25 2019 *)

a(n) = A005316(2n-1) for n>0.

A060206 Number of rotationally symmetric closed meanders of length 4n+2.

Original entry on oeis.org

1, 2, 10, 66, 504, 4210, 37378, 346846, 3328188, 32786630, 329903058, 3377919260, 35095839848, 369192702554, 3925446804750, 42126805350798, 455792943581400, 4967158911871358, 54480174340453578, 600994488311709056, 6664356253639465480

Offset: 0

N. J. A. Sloane, Apr 10 2001

Closed meanders of other lengths do not have rotational symmetry. - Andrew Howroyd, Nov 24 2015

See A077460 for additional information on the symmetries of closed meanders.

R. Bacher, Meander algebras
Andrew Howroyd, Illustration of a(1) and a(2)

Cf. A000682, A005315, A077055, A077460, A223096.

Meander sequences in Bacher's paper: A060066, A060089, A060111, A060148, A060149, A060174, A060198.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := A000682[[2n + 1]];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 03 2019 *)

a(n) = A000682(2n + 1). - Andrew Howroyd, Nov 24 2015

Name edited by Andrew Howroyd, Nov 24 2015

a(7)-a(20) from Andrew Howroyd, Nov 24 2015

A077055 Call two meanders from A005316 equivalent if they differ by a reflection in the Y axis (if n even) or by reflections in the X or Y axes (if n odd). Sequence gives number of inequivalent meanders with n crossings.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 13, 42, 72, 273, 475, 1970, 3506, 15368, 27888, 126510, 233809, 1086546, 2039564, 9652364, 18360296, 88172609, 169610371, 824506191, 1601297937, 7865294687, 15401847339, 76331857094, 150547538649, 751981532942, 1492452957398

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 29 2002

Meander shapes. [Stéphane Legendre, Apr 09 2013]

			For n=7 the A005316(7) = 42 meanders with 7 crossings fall into 5 equivalence classes of size 2 and 8 of size 4, so a(7) = 5+8 = 13.

N. J. A. Sloane, Table of n, a(n) for n = 0..42 [from Legendre, 2013]
CombOS - Combinatorial Object Server, Generate meanders and stamp foldings
Stéphane Legendre, Illustration of initial terms
Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
J. Sawada and R. Li, Stamp foldings, semi-meanders, and open meanders: fast generation algorithms, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).

Cf. A005316, A077460, A078592, A223096.

For n even a(n) = (A005316(n)+A005316(n/2))/2 (this is A078592).

For n odd a(n) = (A005316(n)+2*A223096(floor(n/2)))/4. [Stéphane Legendre, Apr 09 2013]

More terms from the Sawada-Li paper from Daniel Recoskie, Jul 11 2012

A078105 Number of nonisomorphic ways a loop can cross three roads meeting in a Y n times (orbits under symmetry group of order 6).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 8, 8, 48, 54, 331, 439, 2558, 3734, 21057, 33384, 182293, 307719, 1638465, 2913775, 15181584, 28194412, 144206012, 277887666, 1398566992

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 05 2002

There is no constraint on touching any particular sector.

The Mercedes-Benz problem: closed meanders crossing a Y.

			With three crossings the loop must cut each road exactly once, so a(3) = 1.
With 4 crossings the loop can cut one road 4 times (one possibility), or two roads twice each (one possibility), so a(4) = 2.

Anonymous, Illustration for a(3) = 1

Cf. A078104 (total number of solutions), A077460 and A005315 (loop crossing one road).

A078104 Number of ways a loop can cross three roads meeting in a Y n times. The loop must touch the southwest sector.

Original entry on oeis.org

1, 0, 2, 1, 7, 6, 37, 42, 237, 320, 1715, 2610, 13478, 22404, 112480, 200158, 982561, 1846314, 8897089, 17481864

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 05 2002

nonn

The Mercedes-Benz problem: closed meanders crossing a Y.

			With three crossings the loop must cut each road exactly once, so a(3) = 1.
With 4 crossings the loop can cut one road 4 times (giving A005315(2)*2 = 4 possibilities), or two roads twice each (3 ways), so a(4) = 7.

Anonymous, Illustration for a(3) = 1

See A085919 for another version. Cf. A078105 (nonisomorphic solutions), A077460 and A005315 (loop crossing one road).

Cf. also A077550.

More terms added Aug 25 2003

A078591 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

Original entry on oeis.org

1, 1, 1, 4, 21, 131, 914, 6910, 55477, 466729, 4076430, 36712325, 339195058, 3202515525, 30803440806, 301094270964, 2984903334517, 29961600364523, 304094354787062, 3117138919265903, 32238856059792302, 336132907436386486, 3530470987229030696, 37330864330583904876, 397168915877285183906

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 07 2002

Nonisomorphic closed meanders, where two closed meanders are considered equivalent if one can be obtained from the other by reflections in an East-West mirror (a group of order 2).

			A meander can be specified by marking 2n equally spaced points along a line and recording the order in which the meander visits the points.
For n = 2, 4, 6, 8 the solutions are as follows:
n=2: 1 2
n=4: 1 2 3 4
n=6: 1 2 3 4 5 6, 1 2 3 6 5 4, 1 2 5 4 3 6, 1 4 3 2 5 6
n=8: 1 2 3 4 5 6 7 8, 1 2 3 4 5 8 7 6, 1 2 3 4 7 6 5 8, 1 2 7 6 3 4 5 8, 1 2 3 6 7 8 5 4, 1 2 3 6 5 4 7 8,
n=8 (cont.): 1 2 5 4 3 6 7 8, 1 2 3 8 7 6 5 4, 1 2 5 4 3 8 7 6, 1 2 7 6 5 4 3 8, 1 2 3 8 5 6 7 4, 1 2 3 8 7 4 5 6, 1 2 5 6 7 4 3 8,
n=8 (cont.): 1 2 7 4 5 6 3 8, 1 4 3 2 5 6 7 8, 1 4 5 6 3 2 7 8, 1 4 3 2 5 8 7 6, 1 4 3 2 7 6 5 8, 1 6 5 4 3 2 7 8, 1 6 5 2 3 4 7 8, 1 6 3 4 5 2 7 8,

Jean-François Alcover, Table of n, a(n) for n = 0..28

The total number of closed meanders with 2n crossings is given in A005315. Cf. A077055, A078104, A078105, A077460 (same but with group of order 4).

A005315 = Cases[Import["https://oeis.org/A005315/b005315.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n < 3, 1, A005315[[n+1]]/2];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 10 2022, after Andrew Howroyd *)

a(n) = A005315(n) / 2 for n >= 2. - Andrew Howroyd, Nov 23 2015

a(10)-a(20) added by Andrew Howroyd, Nov 23 2015

a(21)-a(28) computed from A005315 added by Jean-François Alcover, Aug 10 2022

cp's OEIS Frontend

A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.

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A060206 Number of rotationally symmetric closed meanders of length 4n+2.

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A077055 Call two meanders from A005316 equivalent if they differ by a reflection in the Y axis (if n even) or by reflections in the X or Y axes (if n odd). Sequence gives number of inequivalent meanders with n crossings.

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A078105 Number of nonisomorphic ways a loop can cross three roads meeting in a Y n times (orbits under symmetry group of order 6).

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A078104 Number of ways a loop can cross three roads meeting in a Y n times. The loop must touch the southwest sector.

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A078591 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

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