A005315 -id:A005315 - 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.

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

Original entry on oeis.org

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

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

A060066 Number of tame meanders with 2n crossings.

Original entry on oeis.org

1, 3, 15, 93, 657, 5063, 41535, 357205, 3187599, 29303687, 276062807, 2654603987

Offset: 1

N. J. A. Sloane, Apr 10 2001

R. Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 26 2001

A060089 Dimensions of graded algebra associated with meanders (subalgebra version).

Original entry on oeis.org

1, 1, 3, 7, 23, 63, 213, 627, 2149, 6597, 22787, 71883, 249523, 802291, 2794365, 9111917, 31814061, 104862813, 366796437, 1219313185, 4271041447, 14295561451, 50131159253, 168742700865, 592279599483, 2003050663889, 7035894016347, 23890177457535, 83968962295531

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with n crossings which are closed on one side and contain no closed loops. These are called unidirectional open meandric systems in the Bobier and Sawada reference. - Andrew Howroyd, Feb 07 2025

Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 0..28 from B. Bobier and J. Sawada)
Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.
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.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 26 2001

Further terms from the Bobier-Sawada paper, Jul 28 2007

A060148 Number of closed forest meander systems with 2n crossings.

Original entry on oeis.org

1, 1, 3, 15, 97, 733, 6147, 55541, 530773, 5298723, 54780831, 582817337, 6350647873, 70614662303, 798935833885, 9176290300419, 106793746090045, 1257408517909283, 14958873368871405, 179614516459970349, 2174717049372338913, 26530091641879493297, 325875790867387681293

Offset: 0

N. J. A. Sloane, Apr 10 2001

A forest meander system is a meander system that does not have any components which are entirely enclosed by another. - Andrew Howroyd, Nov 22 2015

The components of a forest meander system do not necessarily all have exterior arches. See example. Those that do are called tame (A060066). - Andrew Howroyd, Feb 02 2025

			An example of a 2 component forest meander system with 8 crossings that is not tame:
      ________
     / ______ \
    / /      \ \
   / / /\  /\ \ \
   \ \ \/ / /  \/
    \ \__/ /
     \____/

Roland Bacher, Meander algebras

Row sums of A380368.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

1 <= A060066(n) <= a(n) <= A060174(n) <= A060198(n) <= 16^n. - Andrew Howroyd, Feb 02 2025

More terms from Sascha Kurz, Mar 25 2002
a(15)-a(20) from Andrew Howroyd, Nov 22 2015
a(0)=1 prepended and a(21)-a(22) from Andrew Howroyd, Jan 31 2025

A060174 Dimensions of graded algebra associated with forest meanders (subalgebra version).

Original entry on oeis.org

1, 4, 32, 320, 3536, 41344, 501264, 6232736, 78948912, 1014329856, 13178920016, 172789849952, 2282474558160, 30340247738176, 405461900524400, 5443463231234592, 73372619284217776, 992459740253399360, 13465895431077568080, 183211658096891394848, 2498857102607629076880

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with 2n crossings which are closed on one side and whose loops are all even. A loop is even if it has its least point at an even index with crossings numbered from zero. - Andrew Howroyd, Feb 07 2025

Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

Offset corrected and a(7) onwards from Andrew Howroyd, Jan 31 2025

A060198 Dimensions of graded algebra associated with forest meanders.

Original entry on oeis.org

1, 16, 240, 3552, 52224, 764672, 11163936, 162631712, 2365037376, 34344187424, 498139336992, 7217820903328, 104490673015136, 1511512136496064, 21849735526096256, 315654728421607968, 4557598148470097472, 65771755517857808768, 948727279133187672224, 13679121303056997294080, 197153311343380929158240

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with 2n crossings which can be open on both sides and whose loops are all even. A loop is even if it has its least point at an even index with crossings numbered from zero. This definition excludes slices such as (open, open, close, close) since the inner loop is not even. It also excludes (up, open, close, down) even though the loop is not contained in another. - Andrew Howroyd, Feb 07 2025

			a(1) = 240 = 256 - 16 = 4^4 - 4^2 counts all length 4 sequences of open, close, up and down steps, excluding those that have an open + close pair as the central two elements. - _Andrew Howroyd_, Feb 07 2025

Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

Offset corrected and a(7) onwards from Andrew Howroyd, Feb 06 2025

A077054 Number of ways a river can cross a road 2n times.

Original entry on oeis.org

1, 1, 3, 14, 81, 538, 3926, 30694, 252939, 2172830, 19304190, 176343390, 1649008456, 15730575554, 152663683494, 1503962954930, 15012865733351, 151622652413194, 1547365078534578, 15939972379349178, 165597452660771610, 1733609081727968492

Offset: 0

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

nonn

More precisely, number of ways that a river (or directed line) that starts in the southwest and flows east can cross an east-west road 2n times (bisection of A005316).

Also number of ways a loop can cross two parallel roads 2n times. Some portion of loop must lie below lower road.

Jean-François Alcover, Table of n, a(n) for n = 0..27

Bisection of A005316. Cf. A005315, A085873, A086031.

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := A005316[[2n+1]];
a /@ Range[0, 27] (* Jean-François Alcover, Sep 20 2019 *)

a(n) = A005316(2*n).

A060111 Dimensions of graded algebra associated with meanders.

Original entry on oeis.org

1, 4, 15, 56, 207, 764, 2805, 10288, 37609, 137380, 500655, 1823440, 6629423, 24090332, 87418221, 317085352, 1148825185, 4160744164, 15054719697, 54454345624, 196805925995, 711077858188, 2567375653681, 9267176552040, 33430012251123, 120565130387572, 434578910451203

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with n crossings which can be open on both sides and contain no closed loops. These are called open meandric systems in the Bobier and Sawada reference. - Andrew Howroyd, Feb 07 2025

Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 0..27 from B. Bobier and J. Sawada)
Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.
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. [The final term for a(28) is incorrect].

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 26 2001

cp's OEIS Frontend

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

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

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

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A060066 Number of tame meanders with 2n crossings.

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A060089 Dimensions of graded algebra associated with meanders (subalgebra version).

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A060148 Number of closed forest meander systems with 2n crossings.

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A060174 Dimensions of graded algebra associated with forest meanders (subalgebra version).

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A060198 Dimensions of graded algebra associated with forest meanders.

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A077054 Number of ways a river can cross a road 2n times.

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