A005316 Meandric numbers: number of ways a river can cross a road n times. (Formerly M0874)

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

0,4

Comments

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.

References

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).

Links

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).

Crossrefs

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

See also A078592.

Sequence in context: A080877 A007165 A107321 * A076876 A124495 A007919

Adjacent sequences: A005313 A005314 A005315 * A005317 A005318 A005319

Keyword

nonn,nice

Author

N. J. A. Sloane, Stéphane Legendre

Extensions

Computed to n = 43 by Iwan Jensen

Status

approved