Search: a076875 -id:a076875
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A077550
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Number of nonisomorphic ways a river (or undirected line) can cross two perpendicular roads n times (orbits of A076875 under symmetry group of order 8).
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+20
1
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OFFSET
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0,3
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COMMENTS
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There is no constraint on touching any particular sector.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A005316
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Meandric numbers: number of ways a river can cross a road n times.
(Formerly M0874)
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+10
50
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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
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OFFSET
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0,4
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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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.
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)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Computed to n = 43 by Iwan Jensen
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STATUS
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approved
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A076876
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Meandric numbers for a river crossing two parallel roads at n points.
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+10
24
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A076907
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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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+10
24
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2, 2, 6, 10, 32, 62, 210, 436, 1540, 3346, 12192, 27344, 102054, 234388, 891574, 2085940, 8057844, 19134786, 74864648, 179968564, 711708544
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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a(6) and a(7) corrected Aug 25 2003
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STATUS
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approved
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A076906
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Meandric numbers for a river (or directed line) crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in the (+,+) quadrant if n even, in the (+,-) quadrant if n odd.
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+10
23
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0, 1, 2, 5, 12, 31, 82, 218, 612, 1673, 4892, 13672, 41192, 117194, 361302, 1042970, 3274712, 9567393, 30490688, 89984282, 290353456
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OFFSET
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0,3
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more,nice
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AUTHOR
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EXTENSIONS
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a(7) corrected Aug 25 2003
a(7) corrected and a(8)-a(20) added by Robert Price, Jul 29 2012
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STATUS
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approved
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A209656
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Meandric numbers for a river crossing up to 12 parallel roads at n points.
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+10
20
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1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76538, 240368, 665100, 2123379, 5964156, 19301178, 54890366, 179679030
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A204352
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Meandric numbers for a river crossing up to 3 parallel roads at n points.
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+10
19
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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
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A206432
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Meandric numbers for a river crossing any number of parallel roads at n points.
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+10
19
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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
(list;
graph;
refs;
listen;
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text;
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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CROSSREFS
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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).
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A208062
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Meandric numbers for a river crossing up to 4 parallel roads at n points.
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+10
18
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A208126
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Meandric numbers for a river crossing up to 5 parallel roads at n points.
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+10
18
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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