A076906 -id:A076906 - 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.

A076875 Meandric numbers for a curve crossing two perpendicular lines at n points.

Original entry on oeis.org

1, 2, 4, 10, 22, 62, 176, 436

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 24 2002

a(n) = number of ways that a curve can start in the (-,-) quadrant, cross the x and y axes at exactly n points and end in any quadrant. Line is undirected.

			See illustration for a(4)=22: each of the 12 solutions shown crosses the x-axis first and ten of them are related by mirror symmetry to a corresponding curve that crosses the y-axis first, making the total a(4)=22.

Jon Wild, Illustration of a(4) = 22 (ignore the arrowheads)

Cf. A005316, A076876, A076906, A076907 (directed case).

For n odd a(n) = 2*A076906(n).

a(6) and a(7) corrected Aug 23 2003

a(7) corrected by Robert Price, Jul 29 2012

A076876 Meandric numbers for a river crossing two parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 43, 81, 272, 538, 1920, 3926, 14649, 30694, 118489, 252939, 1002994, 2172830, 8805410, 19304190, 79648888, 176343390, 738665040, 1649008456, 6996865599, 15730575554, 67491558466, 152663683494, 661370687363, 1503962954930, 6571177867129

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 26 2002

a(n) = number of ways that a curve can start in the (-,-) quadrant, cross two parallel lines and end up in the (+,+) or (+,-) quadrant if n is even or head East between the two roads if n is odd.
A107321 is a lower bound. - R. J. Mathar, May 06 2006
It appears that for odd n, A076876(n) = A005316(n+1). And for even n, A076876(n) >= A005316(n+1). - Robert Price, Jul 27 2013.

			Let b(n) = A005316(n). Then a(0) = b(0), a(1) = b(1), a(2) = b(1) + b(2), a(3) = b(3) + b(2), a(4) = b(4) + 2*b(3) + 1, a(5) = b(5) + b(3)*b(2) + b(4) + 1.
Consider n=5: if we do not cross the second road there are b(5) = 8 solutions. If we cross the first road 3 times and then the second road twice there are b(3)*b(2) = 2 solutions. If we cross the first road once and the second road 4 times there are b(4) = 3 solutions. The only other possibility is to cross road 1, road 2 twice, road 1 twice and exit to the right.
For larger n it is convenient to give the vector of the number of times the same road is crossed. For example for n=6 the vectors and the numbers of possibilities are as follows:
[6] ...... 14
[5 1] ..... 8
[3 3] ..... 4
[3 2 1] ... 2
[1 5] ..... 8
[1 4 1] ... 3
[1 2 3] ... 2
[1 2 2 1] . 2
Total .... 43

Andrew Howroyd, Table of n, a(n) for n = 0..40
R. J. Mathar, ASCII representations

Column k=2 of A380367.

Cf. A005316, A206432, A204352, A076875, A076906, A076907.

More terms from R. J. Mathar, Mar 04 2007
a(12)-a(20) from Robert Price, Apr 15 2012
a(21)-a(40) from Andrew Howroyd, Dec 07 2015

A076907 Meandric numbers for a river crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in any quadrant.

Original entry on oeis.org

2, 2, 6, 10, 32, 62, 210, 436, 1540, 3346, 12192, 27344, 102054, 234388, 891574, 2085940, 8057844, 19134786, 74864648, 179968564, 711708544

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 26 2002

a(n) = number of ways that a directed curve (or arrow) can start in the (-,-) quadrant, cross the x and y axes at exactly n points and end in any quadrant.

Cf. A005316, A076876, A076906, A076875 (undirected case).

Cf. A077551 (cross x axis first).

a(2n+1) = 2*A076906(2n+1).

a(6) and a(7) corrected Aug 25 2003
a(7) corrected by Robert Price, Jul 29 2012
a(8)-a(20) from Robert Price, Aug 01 2012

A209656 Meandric numbers for a river crossing up to 12 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76538, 240368, 665100, 2123379, 5964156, 19301178, 54890366, 179679030, 516360755, 1706896545, 4949350203, 16500278295, 48216373545, 161946759019, 476447428528, 1610847688579, 4767486352733, 16213635060406

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 12 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=12 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A204352 Meandric numbers for a river crossing up to 3 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 52, 131, 345, 915, 2519, 6926, 19711, 55674, 162594, 468929, 1398129, 4100003, 12433282, 36960316, 113678461, 341785050, 1063890616, 3229522688, 10156518859, 31085477306, 98635931623, 304048850048, 972323924567, 3015979607106, 9711570455824

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross 3 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=3 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432  (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21)-a(40) from Andrew Howroyd, Dec 07 2015

A206432 Meandric numbers for a river crossing any number of parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898417, 179696559, 516468945, 1707136837, 4950706599, 16503343162, 48232630706, 161984048816, 476636485050, 1611287098347, 4769639932874, 16218699278307

Offset: 0

Robert Price, May 07 2012

Number of ways that a river (or directed line) that starts in the South and flows East can cross any number of parallel East-West roads n times.
Of course, the number of roads crossed cannot be more than the number of crossings, n.
A file (28GB) listing all solutions through n=20 is available from the author.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A076875, A076906, A076907.

a(21)-a(30) from Andrew Howroyd, Dec 07 2015

A208062 Meandric numbers for a river crossing up to 4 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 61, 142, 420, 1017, 3146, 7844, 25083, 63974, 209875, 545060, 1824949, 4810138, 16374993, 43695535, 150820899, 406669871, 1420155120, 3863613980, 13627843933, 37363313071, 132933980698, 366939582498, 1315436809855, 3652777067949, 13182411646150

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 4 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..36

Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432  (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21)-a(36) from Andrew Howroyd, Dec 05 2015

A208126 Meandric numbers for a river crossing up to 5 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 62, 153, 433, 1120, 3281, 8776, 26399, 72423, 222496, 622616, 1946044, 5533227, 17545134, 50545069, 162237507, 472541542, 1532707268, 4506042037, 14748997812, 43709690021, 144213436887, 430371626111, 1429980808522, 4293569179189, 14355812667699

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 5 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..36

Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21)-a(36) from Andrew Howroyd, Dec 05 2015

A208452 Meandric numbers for a river crossing up to 6 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 154, 446, 1135, 3416, 8947, 27712, 74204, 235041, 640552, 2065849, 5711840, 18696838, 52321747, 173413647, 490279035, 1642287831, 4684178185, 15834649146, 45511151647, 155077912223, 448722569787, 1539743907074, 4481866870316, 15474663284941

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 6 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..36

Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21)-a(36) from Andrew Howroyd, Dec 05 2015

cp's OEIS Frontend

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

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A076875 Meandric numbers for a curve crossing two perpendicular lines at n points.

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A076876 Meandric numbers for a river crossing two parallel roads at n points.

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A076907 Meandric numbers for a river crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in any quadrant.

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A209656 Meandric numbers for a river crossing up to 12 parallel roads at n points.

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A204352 Meandric numbers for a river crossing up to 3 parallel roads at n points.

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A206432 Meandric numbers for a river crossing any number of parallel roads at n points.

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A208062 Meandric numbers for a river crossing up to 4 parallel roads at n points.

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A208126 Meandric numbers for a river crossing up to 5 parallel roads at n points.

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A208452 Meandric numbers for a river crossing up to 6 parallel roads at n points.

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