A005316 -id:A005316 - cp's OEIS frontend

A006658 Closed meanders with 3 components and 2n bridges.

5, 56, 580, 5894, 60312, 624240, 6540510, 69323910, 742518832, 8028001566, 87526544560, 961412790002, 10630964761766, 118257400015312, 1322564193698320, 14863191405246888, 167771227744292160, 1901345329566422790

Offset: 3

N. J. A. Sloane

nonn

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
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)

Cf. A005315, A006657.

A column of triangle A008828.

A008828 = Import["https://oeis.org/A008828/b008828.txt", "Table"][[All, 2]];
a[n_] := A008828[[(n^2 - n + 6)/2]];
a /@ Range[3, 20] (* Jean-François Alcover, Sep 25 2019 *)

a(13)-a(20) from Andrew Howroyd, Nov 22 2015

A006661 Number of meanders in which first bridge is 5.

Original entry on oeis.org

3, 3, 7, 11, 28, 57, 155, 353, 1003, 2458, 7214, 18575, 55880, 149183, 457639, 1255933, 3914103, 10978240, 34663182, 98953078, 315884786, 915008430, 2948378068, 8645874055, 28084475514, 83222134020

Offset: 5

Simon Plouffe

nonn

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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)

Cf. A259974, A005316.

Offset corrected terms a(19)-a(30) added, Andrew Howroyd, Dec 15 2015

A006662 Number of meanders in which first bridge is 7.

Original entry on oeis.org

14, 14, 36, 57, 155, 316, 902, 2053, 6059, 14810, 44842, 115009, 355293, 943860, 2963536, 8086913, 25733325, 71725012, 230811370, 654472364, 2126296860, 6115504594, 20032488714, 58309793101

Offset: 7

Simon Plouffe

nonn

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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)

Cf. A259974, A005316.

Offset corrected, terms a(19)-a(30) added, Andrew Howroyd, Dec 15 2015

A085973 Number of ways a loop can cross two parallel roads 2n times.

Original entry on oeis.org

3, 2, 5, 22, 123, 800, 5754, 44514, 363893, 3106288, 27457050, 249768040, 2327398572, 22135606604, 214270565106, 2106151496858, 20982672402385, 211545853142240, 2155553788108702, 22174250217880984, 230075164780356214

Offset: 0

N. J. A. Sloane and Jon Wild, Aug 25 2003

nonn

There is no obligation to cross the lower road (cf. A077054).

Cf. A077054, A086031.

A005315 = Cases[Import["https://oeis.org/A005315/b005315.txt", "Table"], {, }][[All, 2]];
A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 3, A005315[[n + 1]] + A005316[[2 n + 1]]];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 08 2019 *)

a(n) = A077054(n) + A005315(n) for n >= 1. - Andrew Howroyd, Nov 26 2015

a(13)-a(20) from Andrew Howroyd, Nov 26 2015

A227167 The number of meandering curves of order n.

Original entry on oeis.org

1, 1, 6, 8, 50, 72, 462, 696, 4536, 7030, 46310, 73188, 485914, 778946, 5202690, 8430992, 56579196, 92470194, 622945970, 1025114180, 6927964218, 11465054942, 77692142980, 129180293184, 877395996200, 1464716085664, 9968202968958, 16698145444260, 113837957337750, 191264779292430

Offset: 1

Antonios Panayotopoulos, Panayotis Vlamos, Jul 03 2013

nonn

A meandering curve of order n is a continuous curve which does not intersect itself yet intersects a horizontal line n times.

The set of meandering curves of order n is partitioned into the following three classes: curves with no extremities (A005316), curves with only one extremity (A217310), and curves with both extremities covered by their arcs (A217318).

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

Jean-François Alcover, Table of n, a(n) for n = 1..45
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152.
W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193-199.
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.

A000136 = Cases[Import["https://oeis.org/A000136/b000136.txt", "Table"], {, }][[All, 2]];
a[n_] := If[OddQ[n], A000136[[n]], A000136[[n]]/2];
a /@ Range[1, 45] (* Jean-François Alcover, Sep 20 2019 *)

a(n) = A000136(n) if n is odd and a(n) = (1/2)*A000136(n) if n is even.

a(n) = A217310(n) + A217318(n) + A005316(n). - Andrew Howroyd, Dec 07 2015

A230439 Number of contractible "tight" meanders of width n.

Original entry on oeis.org

1, 2, 6, 14, 34, 68, 150, 296, 586, 1140, 2182, 4130, 7678, 14368, 26068, 48248, 86572, 158146, 281410, 509442, 901014, 1618544, 2852464, 5089580, 8948694, 15884762, 27882762, 49291952, 86435358, 152316976, 266907560, 469232204, 821844316

Offset: 1

Mamuka Jibladze, Nov 04 2013

nonn

A tight meander of width n is a special kind of meander defined as follows.

For any pair (S={s_1,...,s_k},T={t_1,...,t_l}) of subsets of {1,...,n-1} (k or l might be 0), the tight meander M(S,T) defined by (S,T) is the following subset of R^2:

assuming S and T ordered so that 0=s_0

semicircles in the upper half-plane with endpoints (s_{i-1}+j,0) and (s_i+1-j,0), for i=1,...,k+1, and j positive integer with s_{i-1}+j

and semicircles in the lower half-plane with endpoints (t_{i-1}+j,0) and (t_i+1-j,0), for i=1,...,l+1, and j positive integer with t_{i-1}+j

The tight meander M(S,T) is called contractible if it is a contractible subspace of R^2, i.e., is either a single point or homeomorphic to an interval.

Then, a(n) is the number of pairs (S,T) as above such that the tight meander M(S,T) is contractible.

From Roger Ford, Jul 05 2023: (Start)

The following is a definition for closed meanders that yield the same sequence as tight meanders. T(n,k) = the number of closed meanders with n top arches and with k exterior arches and k arches of length 1.

e = exterior arch (arch with no covering arch), 1 = arch with length 1, e1 = arch that is exterior with a length of 1:

e exterior length 1

__________ arches arches

/ ____ \

e1 / / \ \ top = 2 top = 2

/\ / / /\1 \ \

/ \ / / / \ \ \

\ \ / / \ \ / / bottom = 2 bottom = 2

\ \/1 / \ \/1 / total = 4 total = 4

\____/ \____/

e e Example T(4,4).

(End)

			For n=3 the a(3)=6 contractible tight meanders of width 3 correspond to the following pairs of subsets of {1,2}: ({},{1}), ({},{2}), ({1},{}), ({2},{}), ({1},{2}), ({2},{1}).

Mamuka Jibladze, Table of n, a(n) for n = 1..100 (first 64 terms by Martin Plechsmid)
Vincent Coll, Colton Magnant, and Hua Wang, The Signature of a Meander, arXiv:1206.2705 [math.QA], 2012.
Vladimir Dergachev and Alexandre Kirillov, Index of Lie algebras of Seaweed Type, J. Lie Theory, 10 (2000), 331-343
Mathoverflow, "Special" meanders
Dmitri I. Panyushev, Inductive Formulas for the Index of Seaweed Lie Algebras, Moscow Math. J., 1 (2001), 221-241.

For various kinds of meandric numbers see A005315, A005316, A060066, A060089, A060206.

# program based on the C code by Martin Plechsmid:
proc()
local n,a,b,d,r;
option remember;
  if args[1]=1 then
   1
  elif nargs=1 then
   2*`+`(''procname(args,[i],[j])'$'j'=1..i-1'$'i'=2..args)
  else
   n:=args[1]; a:=args[2]; b:=args[3];
   if b=[] then
    `+`('procname(n,a,[k])'$'k'=1..n)
   elif a[1]=b[1] then
    0
   elif a[1]0 then
     procname(n-b[1],[d-r,op(subsop(1=r,a))],subsop(1=NULL,b))
    else
     procname(n-b[1],subsop(1=d,a),subsop(1=NULL,b))
    fi
   fi
  fi
end;

(* program based on the C code by Martin Plechsmid: *)
f[n_,a_,b_]:=Which[
n==1, 1,
b=={}, f[n,a,b]=Sum[f[n,a,{i}],{i,n}],
a=={} || First[a]

A368054 Irregular triangle read by rows: T(n,k) is the number of k-crossing partitions on 2n nodes, where all partition terms alternate in parity, counted up to reflection.

Original entry on oeis.org

1, 1, 3, 0, 1, 14, 0, 8, 10, 2, 2, 81, 0, 59, 162, 70, 66, 82, 22, 19, 6, 7, 0, 2, 538, 0, 454, 1952, 1229, 1208, 2516, 1803, 1181, 1148, 998, 478, 370, 279, 125, 76, 26, 13, 3, 3, 3926, 0, 3658, 21608, 17083, 17811, 48542, 51306, 40081, 51660, 59023, 42327

Offset: 0

John Tyler Rascoe, Dec 09 2023

The 0-crossing partitions counted in A005316 all have terms that alternate in parity. Also, for an even number of nodes the partitions 1432 and 2341 count the same meandric path. This triangle aims to reduce the total number of k-crossing partitions considered from (2*n)! to (n!)^2, see Irwin link.

			Triangle begins:
       k=0  1   2    3   4   5   6   7   8   9  10  11  12
  n=0:   1;
  n=1:   1;
  n=2:   3, 0,  1;
  n=3:  14, 0,  8,  10,  2,  2;
  n=4:  81, 0, 59, 162, 70, 66, 82, 22, 19,  6,  7,  0,  2;
  ...
Row n = 3 counts the following k-crossing partitions.
T(3,0) = 14:   T(3,2) = 8:    T(3,3) = 10:   T(3,4) = 2:    T(3,5) = 2:
(1,2,3,4,5,6)  (3,4,1,6,5,2)  (1,2,5,6,3,4)  (3,2,5,6,1,4)  (3,6,1,4,5,2)
(1,2,3,6,5,4)  (3,4,5,6,1,2)  (1,4,3,6,5,2)  (3,6,1,2,5,4)  (5,2,3,6,1,4)
(1,2,5,4,3,6)  (3,6,5,4,1,2)  (1,4,5,2,3,6)
(1,4,3,2,5,6)  (5,2,1,6,3,4)  (1,6,3,2,5,4)
(1,4,5,6,3,2)  (5,4,3,6,1,2)  (3,2,5,4,1,6)
(1,6,3,4,5,2)  (5,6,1,2,3,4)  (3,4,1,2,5,6)
(1,6,5,2,3,4)  (5,6,1,4,3,2)  (3,6,5,2,1,4)
(1,6,5,4,3,2)  (5,6,3,2,1,4)  (5,2,1,4,3,6)
(3,2,1,4,5,6)                 (5,4,1,6,3,2)
(3,2,1,6,5,4)                 (5,6,3,4,1,2)
(3,4,5,2,1,6)
(5,2,3,4,1,6)
(5,4,1,2,3,6)
(5,4,3,2,1,6)

Benedict Irwin, On the Number of k-Crossing Partitions, Univ. of Cambridge (2021).
John Tyler Rascoe, Python program.

Cf. A077054 (column k=0), A001044 (row sums).

Cf. A005316, A008828, A287220.

Python
```
# see linked program
```

A006663 Number of projective meanders.

Original entry on oeis.org

1, 1, 2, 2, 8, 12, 52, 86, 400, 710, 3404, 6316, 30888, 59204, 293192, 576018, 2877184, 5764430, 28967428, 58970568, 297634344, 614037754, 3109111064

Offset: 0

N. J. A. Sloane, Simon Plouffe

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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 (1993), no. 1-2, 227-241.

a(16)-a(22) from Andrew Howroyd, Mar 30 2017

A167512 Number of Simple Alternating Transit (SAT) mazes with exactly two extreme values.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 10, 22, 34, 67, 100, 188

Offset: 0

Kristen M. Hendershot (KHendershot(AT)westliberty.edu), Nov 05 2009

K(N) incorporates Fibonacci numbers to find the solution.

Tony Phillips, Phillips Mazes

Cf. A000045, A005316

A287222 Number of 3-time self-crossing partitions on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 2, 16, 164, 944, 4386, 22240, 83066, 398132

Offset: 0

Benedict W. J. Irwin, May 22 2017

The meandric numbers A005316 are the numbers of paths which cross themselves 0 times.

This sequence is the number of paths that must cross themselves exactly 3 times.

			a(4) = 2, this is from the partitions (2,4,1,3) and (3,4,1,2).

B. W. J. Irwin, On the number of k-crossing partitions

Cf. A005316, A287220, A287221.

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cp's OEIS Frontend

A006658 Closed meanders with 3 components and 2n bridges.

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A006661 Number of meanders in which first bridge is 5.

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A006662 Number of meanders in which first bridge is 7.

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A085973 Number of ways a loop can cross two parallel roads 2n times.

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A227167 The number of meandering curves of order n.

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A230439 Number of contractible "tight" meanders of width n.

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Maple

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A368054 Irregular triangle read by rows: T(n,k) is the number of k-crossing partitions on 2n nodes, where all partition terms alternate in parity, counted up to reflection.

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Programs

Python

A006663 Number of projective meanders.

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Extensions

A167512 Number of Simple Alternating Transit (SAT) mazes with exactly two extreme values.

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