A077055 -id:A077055 - cp's OEIS frontend

A005316 Meandric numbers: number of ways a river can cross a road n times.

1, 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, 3926, 13820, 30694, 110954, 252939, 933458, 2172830, 8152860, 19304190, 73424650, 176343390, 678390116, 1649008456, 6405031050, 15730575554, 61606881612, 152663683494, 602188541928, 1503962954930, 5969806669034, 15012865733351, 59923200729046, 151622652413194, 608188709574124, 1547365078534578, 6234277838531806, 15939972379349178, 64477712119584604, 165597452660771610, 672265814872772972, 1733609081727968492, 7060941974458061392

Offset: 0

N. J. A. Sloane, Stéphane Legendre

Number of ways that a river (or directed line) that starts in the southwest and flows east can cross an east-west road n times (see the illustration).

Or, number of ways that an undirected line can cross a road with at least one end below the road.

Alon, Noga and Maass, Wolfgang, Meanders and their applications in lower bounds arguments. Twenty-Seventh Annual IEEE Symposium on the Foundations of Computer Science (Toronto, ON, 1986). J. Comput. System Sci. 37 (1988), no. 2, 118-129.
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.
V. I. Arnol'd, ed., Arnold's Problems, Springer, 2005; Problem 1989-18.
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, ACM Transactions on Algorithms, Vol. 6, No. 2, 2010, article #42.
Di Francesco, P. The meander determinant and its generalizations. Calogero-Moser-Sutherland models (Montreal, QC, 1997), 127-144, CRM Ser. Math. Phys., Springer, New York, 2000.
Di Francesco, P., SU(N) meander determinants. J. Math. Phys. 38 (1997), no. 11, 5905-5943.
Di Francesco, P. Truncated meanders. Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), 135-162, Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999.
Di Francesco, P. Meander determinants. Comm. Math. Phys. 191 (1998), no. 3, 543-583.
Di Francesco, P. Exact asymptotics of meander numbers. Formal power series and algebraic combinatorics (Moscow, 2000), 3-14, Springer, Berlin, 2000.
Di Francesco, P., Golinelli, O. and Guitter, E., Meanders. In The Mathematical Beauty of Physics (Saclay, 1996), pp. 12-50, Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, 1997.
Di Francesco, P., Golinelli, O. and Guitter, E. Meanders and the Temperley-Lieb algebra. Comm. Math. Phys. 186 (1997), no. 1, 1-59.
Di Francesco, P., Guitter, E. and Jacobsen, J. L. Exact meander asymptotics: a numerical check. Nuclear Phys. B 580 (2000), no. 3, 757-795.
Franz, Reinhard O. W. A partial order for the set of meanders. Ann. Comb. 2 (1998), no. 1, 7-18.
Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.
Isakov, N. M. and Yarmolenko, V. I. Bounded meander approximations. (Russian) Qualitative and approximate methods for the investigation of operator equations (Russian), 71-76, 162, Yaroslav. Gos. Univ., 1981.
Lando, S. K. and Zvonkin, A. K. Plane and projective meanders. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).
Lando, S. K. and Zvonkin, A. K. Meanders. In Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 117-144.
Makeenko, Y., Strings, matrix models and meanders. Theory of elementary particles (Buckow, 1995). Nuclear Phys. B Proc. Suppl. 49 (1996), 226-237.
A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018). https://doi.org/10.1007/s11786-018-0389-6.
A. Phillips, Simple Alternating Transit Mazes, unpublished. 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.
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).

Andrew Howroyd, Table of n, a(n) for n = 0..55 (first 44 terms from Iwan Jensen)
V. I. Arnold, Problem: continue the sequence 1, 1, 2, 3, 8, 14, 42, 81..., Manuscript.
David Bevan, Open Meanders [From _David Bevan_, Jun 25 2010]
CombOS - Combinatorial Object Server, Generate meanders and stamp foldings
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995; Combinatorics and physics (Marseilles, 1995). Math. Comput. Modelling 26 (1997), no. 8-10, 97-147.
Di Francesco, P., Golinelli, O. and Guitter, E., Meanders: exact asymptotics, Nuclear Phys. B 570 (2000), no. 3, 699-712.
Di Francesco, P., Golinelli, O. and Guitter, E., Meanders: a direct enumeration approach, arXiv:cond-mat/9910453 [cond-mat.stat-mech], 1999-2000; Nuclear Phys. B 482 (1996), no. 3, 497-535.
Andrew Howroyd, C# Software for the enumeration of meanders
Benedict Irwin, On the Number of k-Crossing Partitions, Univ. of Cambridge (2021).
I. Jensen, Home page
I. Jensen, A transfer matrix approach to the enumeration of plane meanders, arXiv:cond-mat/0008178 [cond-mat.stat-mech], 2000.
I. Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
I. Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
I. Jensen, Open meanders, a(n) for n = 0..43
I. Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).
M. La Croix, Approaches to the Enumerative Theory of Meanders [From _Gerald McGarvey_, Oct 26 2008]
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.
S. Legendre, Foldings and Meanders, Aust. J. Comb. 58(2), 275-291, 2014; and also on arXiv, arXiv:1302.2025 [math.CO], 2013.
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.
A. Phillips, Mazes
A. Phillips, Simple, Alternating, Transit Mazes
Frank Ruskey, Information on Stamp Foldings
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).
M. Skrzypczak and P. Pokorski, Illustration of a(10)
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

a(2n) is A005315. Cf. A076875, A076906, A076907, A077014, A077054, A077055, A077056, A078591.

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

A077460 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

Original entry on oeis.org

1, 1, 1, 3, 12, 70, 464, 3482, 27779, 233556, 2038484, 18357672, 169599492, 1601270562, 15401735750, 150547249932, 1492451793728, 14980801247673, 152047178479946, 1558569469867824, 16119428039548246

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 03 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 or North-South mirror (a group of order 4).

Symmetries are possible by reflection in a North-South mirror, or by rotation through 180 degrees when n is odd.(see illustration). - Andrew Howroyd, Nov 24 2015

			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
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, 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, 1 2 7 4 5 6 3 8, 1 4 3 2 7 6 5 8

Andrew Howroyd, Illustration of Closed Meander Symmetries

The total number of closed meanders with 2n crossings is given in A005315. Cf. A000682, A005316, A060206, A077055, A078104, A078105, A078591.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[0] = a[1] = 1;
a[n_] := If[OddQ[n], (A005316[[n + 1]] + A005316[[2n]] + A000682[[n]])/4, (A005316[[2n]] + 2 A005316[[n + 1]])/4];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 06 2019, after Andrew Howroyd *)

From Andrew Howroyd, Nov 24 2015: (Start)
a(2n+1) = (A005315(2n+1) + A005316(2n+1) + A060206(n)) / 4.
a(2n) = (A005315(2n) + 2 * A005316(2n)) / 4. (End)

a(10)-a(20) from Andrew Howroyd, Nov 24 2015

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

A005316 Meandric numbers: number of ways a river can cross a road n times.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A077460 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A223096 Number of symmetric meander shapes with 2n+1 crossings.

Views

Author

Keywords

Comments

Links

Crossrefs