A077014 -id:A077014 - 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.

A223093 Number of foldings of n labeled stamps in which leaf 1 is inwards and leaf n outwards (or leaf 1 outwards and leaf n inwards).

Original entry on oeis.org

0, 0, 2, 4, 16, 38, 132, 342, 1144, 3134, 10370, 29526, 97458, 285458, 942920, 2822310, 9341008, 28440970, 94358558, 291294678, 968853072, 3025232480, 10086634316, 31797822936, 106265437078, 337731551446, 1131117792978, 3620119437762, 12148796744234, 39118879440938

Offset: 1

N. J. A. Sloane, Mar 29 2013

nonn

Subset of foldings of n labeled stamps (A000136). - Stéphane Legendre, Apr 09 2013

Stéphane Legendre, Table of n, a(n) for n = 1..42
S. Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
S. Legendre, Foldings and Meanders, Aust. J. Comb. 58(2), 275-291, 2014.
Index entries for sequences obtained by enumerating foldings

A000682 = Cases[Import["https://oeis.org/A000682/b000682.txt", "Table"], {, }][[All, 2]];
A077014 = Cases[Import["https://oeis.org/A077014/b077014.txt", "Table"], {, }][[All, 2]];
a[n_] := A000682[[n + 1]] - A077014[[n + 1]];
Array[a, 30] (* Jean-François Alcover, Sep 07 2019, after Andrew Howroyd *)

a(n) = A000682(n+1) - A077014(n). - Andrew Howroyd, Dec 06 2015

A217310(n) = 2*a(n) if n is odd and A217310(n) = a(n) if n is even. - Stéphane Legendre, Jan 09 2014

Name clarified by Stéphane Legendre, Apr 09 2013

More terms from Stéphane Legendre, Apr 09 2013

A223095 Number of foldings of n labeled stamps in which both end leaves are inwards.

Original entry on oeis.org

0, 0, 0, 2, 10, 40, 156, 546, 1986, 6716, 23742, 79472, 277178, 925588, 3205896, 10711486, 36963722, 123712788, 426075994, 1429030624, 4916833424, 16526958144, 56840484232, 191466923584, 658460090994, 2222507917328, 7644360501390, 25850724646008, 88938175307354

Offset: 1

N. J. A. Sloane, Mar 29 2013

nonn

Subset of foldings of n labeled stamps (A000136). - Stéphane Legendre, Apr 09 2013

Stéphane Legendre, Table of n, a(n) for n = 1..42
S. Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
S. Legendre, Foldings and Meanders, Aust. J. Comb. 58(2), 275-291, 2014.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A000682, A077014.

Cf. A223093, A223094.

A217318 = Import["https://oeis.org/A217318/b217318.txt", "Table"][[All, 2]];
a[n_] := (2 - Mod[n, 2]) A217318[[n]];
Array[a, 42] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A223094(n) - A223093(n). - Andrew Howroyd, Dec 06 2015
a(n) = A000136(n) + A077014(n) - 2 * A000682(n). - Andrew Howroyd, Dec 06 2015
A217318(n) = a(n) if n is odd and A217318(n) = (1/2)*a(n) if n is even. - Stéphane Legendre, Jan 13 2014

Name clarified by Stéphane Legendre, Apr 09 2013

More terms from Stéphane Legendre, Apr 09 2013

cp's OEIS Frontend

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

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A223093 Number of foldings of n labeled stamps in which leaf 1 is inwards and leaf n outwards (or leaf 1 outwards and leaf n inwards).

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A223095 Number of foldings of n labeled stamps in which both end leaves are inwards.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions