%I M0874 #117 Oct 29 2023 21:50:37
%S 1,1,1,2,3,8,14,42,81,262,538,1828,3926,13820,30694,110954,252939,
%T 933458,2172830,8152860,19304190,73424650,176343390,678390116,
%U 1649008456,6405031050,15730575554,61606881612,152663683494,602188541928,1503962954930,5969806669034,15012865733351,59923200729046,151622652413194,608188709574124,1547365078534578,6234277838531806,15939972379349178,64477712119584604,165597452660771610,672265814872772972,1733609081727968492,7060941974458061392
%N Meandric numbers: number of ways a river can cross a road n times.
%C 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).
%C Or, number of ways that an undirected line can cross a road with at least one end below the road.
%D 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.
%D 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.
%D V. I. Arnol'd, ed., Arnold's Problems, Springer, 2005; Problem 1989-18.
%D 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.
%D 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.
%D Di Francesco, P., SU(N) meander determinants. J. Math. Phys. 38 (1997), no. 11, 5905-5943.
%D 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.
%D Di Francesco, P. Meander determinants. Comm. Math. Phys. 191 (1998), no. 3, 543-583.
%D Di Francesco, P. Exact asymptotics of meander numbers. Formal power series and algebraic combinatorics (Moscow, 2000), 3-14, Springer, Berlin, 2000.
%D 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.
%D Di Francesco, P., Golinelli, O. and Guitter, E. Meanders and the Temperley-Lieb algebra. Comm. Math. Phys. 186 (1997), no. 1, 1-59.
%D Di Francesco, P., Guitter, E. and Jacobsen, J. L. Exact meander asymptotics: a numerical check. Nuclear Phys. B 580 (2000), no. 3, 757-795.
%D Franz, Reinhard O. W. A partial order for the set of meanders. Ann. Comb. 2 (1998), no. 1, 7-18.
%D Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.
%D 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.
%D Lando, S. K. and Zvonkin, A. K. Plane and projective meanders. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).
%D Lando, S. K. and Zvonkin, A. K. Meanders. In Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 117-144.
%D Makeenko, Y., Strings, matrix models and meanders. Theory of elementary particles (Buckow, 1995). Nuclear Phys. B Proc. Suppl. 49 (1996), 226-237.
%D A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018). https://doi.org/10.1007/s11786-018-0389-6.
%D 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.
%D J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A005316/b005316.txt">Table of n, a(n) for n = 0..55</a> (first 44 terms from Iwan Jensen)
%H V. I. Arnold, <a href="/A005316/a005316.pdf">Problem: continue the sequence 1, 1, 2, 3, 8, 14, 42, 81...</a>, Manuscript.
%H David Bevan, <a href="http://demonstrations.wolfram.com/OpenMeanders">Open Meanders</a> [From _David Bevan_, Jun 25 2010]
%H CombOS - Combinatorial Object Server, <a href="http://combos.org/meander">Generate meanders and stamp foldings</a>
%H P. Di Francesco, O. Golinelli and E. Guitter, <a href="http://arXiv.org/abs/hep-th/9506030">Meander, folding and arch statistics</a>, arXiv:hep-th/9506030, 1995; Combinatorics and physics (Marseilles, 1995). Math. Comput. Modelling 26 (1997), no. 8-10, 97-147.
%H Di Francesco, P., Golinelli, O. and Guitter, E., <a href="http://arXiv.org/abs/cond-mat/9910453">Meanders: exact asymptotics</a>, Nuclear Phys. B 570 (2000), no. 3, 699-712.
%H Di Francesco, P., Golinelli, O. and Guitter, E., <a href="http://arXiv.org/abs/hep-th/9607039">Meanders: a direct enumeration approach</a>, arXiv:cond-mat/9910453 [cond-mat.stat-mech], 1999-2000; Nuclear Phys. B 482 (1996), no. 3, 497-535.
%H Andrew Howroyd, <a href="/A005316/a005316.cs.txt">C# Software for the enumeration of meanders</a>
%H Benedict Irwin, <a href="https://doi.org/10.22541/au.162022396.68662845/v1">On the Number of k-Crossing Partitions</a>, Univ. of Cambridge (2021).
%H I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/">Home page</a>
%H I. Jensen, <a href="http://arXiv.org/abs/cond-mat/0008178">A transfer matrix approach to the enumeration of plane meanders</a>, arXiv:cond-mat/0008178 [cond-mat.stat-mech], 2000.
%H I. Jensen, <a href="http://dx.doi.org/10.1088/0305-4470/33/34/301">A transfer matrix approach to the enumeration of plane meanders</a>, J. Phys. A 33, 5953-5963 (2000).
%H I. Jensen, <a href="http://arxiv.org/abs/cond-mat/9910313">Enumeration of plane meanders</a>, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
%H I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/meanders/series/open.meanders.ser">Open meanders, a(n) for n = 0..43</a>
%H I. Jensen and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/33/21/101">Critical exponents of plane meanders</a> J. Phys. A 33, L187-L192 (2000).
%H M. La Croix, <a href="http://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf">Approaches to the Enumerative Theory of Meanders</a> [From _Gerald McGarvey_, Oct 26 2008]
%H S. K. Lando and A. K. Zvonkin, <a href="/A005316/a005316_1.pdf">Plane and projective meanders</a>, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
%H S. K. Lando and A. K. Zvonkin, <a href="http://dx.doi.org/10.1016/0304-3975(93)90316-L">Plane and projective meanders</a>, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
%H S. Legendre, <a href="http://ajc.maths.uq.edu.au/pdf/58/ajc_v58_p275.pdf">Foldings and Meanders</a>, Aust. J. Comb. 58(2), 275-291, 2014; and also <a href="https://arxiv.org/abs/1302.2025">on arXiv</a>, arXiv:1302.2025 [math.CO], 2013.
%H A. Panayotopoulos, P. Vlamos, <a href="https://doi.org/10.1007/s11786-015-0234-0">Partitioning the Meandering Curves</a>, Mathematics in Computer Science (2015) p 1-10.
%H A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/index.html">Mazes</a>
%H A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/satmaze.html">Simple, Alternating, Transit Mazes</a>
%H Frank Ruskey, <a href="http://combos.org/meander">Information on Stamp Foldings</a>
%H J. Sawada and R. Li, <a href="https://doi.org/10.37236/2404">Stamp foldings, semi-meanders, and open meanders: fast generation algorithms</a>, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).
%H M. Skrzypczak and P. Pokorski, <a href="/A005316/a005316_2.pdf">Illustration of a(10)</a>
%H N. J. A. Sloane, <a href="/A005316/a005316.jpg">Illustration of initial terms</a>
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%Y a(2n) is A005315. Cf. A076875, A076906, A076907, A077014, A077054, A077055, A077056, A078591.
%Y See also A078592.
%K nonn,nice
%O 0,4
%A _N. J. A. Sloane_, _Stéphane Legendre_
%E Computed to n = 43 by Iwan Jensen