A060174 -id:A060174 - cp's OEIS frontend

A060206 Number of rotationally symmetric closed meanders of length 4n+2.

1, 2, 10, 66, 504, 4210, 37378, 346846, 3328188, 32786630, 329903058, 3377919260, 35095839848, 369192702554, 3925446804750, 42126805350798, 455792943581400, 4967158911871358, 54480174340453578, 600994488311709056, 6664356253639465480

Offset: 0

N. J. A. Sloane, Apr 10 2001

Closed meanders of other lengths do not have rotational symmetry. - Andrew Howroyd, Nov 24 2015

See A077460 for additional information on the symmetries of closed meanders.

R. Bacher, Meander algebras
Andrew Howroyd, Illustration of a(1) and a(2)

Cf. A000682, A005315, A077055, A077460, A223096.

Meander sequences in Bacher's paper: A060066, A060089, A060111, A060148, A060149, A060174, A060198.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := A000682[[2n + 1]];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 03 2019 *)

a(n) = A000682(2n + 1). - Andrew Howroyd, Nov 24 2015

Name edited by Andrew Howroyd, Nov 24 2015

a(7)-a(20) from Andrew Howroyd, Nov 24 2015

A060066 Number of tame meanders with 2n crossings.

Original entry on oeis.org

1, 3, 15, 93, 657, 5063, 41535, 357205, 3187599, 29303687, 276062807, 2654603987

Offset: 1

N. J. A. Sloane, Apr 10 2001

R. Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 26 2001

A060089 Dimensions of graded algebra associated with meanders (subalgebra version).

Original entry on oeis.org

1, 1, 3, 7, 23, 63, 213, 627, 2149, 6597, 22787, 71883, 249523, 802291, 2794365, 9111917, 31814061, 104862813, 366796437, 1219313185, 4271041447, 14295561451, 50131159253, 168742700865, 592279599483, 2003050663889, 7035894016347, 23890177457535, 83968962295531

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with n crossings which are closed on one side and contain no closed loops. These are called unidirectional open meandric systems in the Bobier and Sawada reference. - Andrew Howroyd, Feb 07 2025

Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 0..28 from B. Bobier and J. Sawada)
Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 26 2001

Further terms from the Bobier-Sawada paper, Jul 28 2007

A060148 Number of closed forest meander systems with 2n crossings.

Original entry on oeis.org

1, 1, 3, 15, 97, 733, 6147, 55541, 530773, 5298723, 54780831, 582817337, 6350647873, 70614662303, 798935833885, 9176290300419, 106793746090045, 1257408517909283, 14958873368871405, 179614516459970349, 2174717049372338913, 26530091641879493297, 325875790867387681293

Offset: 0

N. J. A. Sloane, Apr 10 2001

A forest meander system is a meander system that does not have any components which are entirely enclosed by another. - Andrew Howroyd, Nov 22 2015

The components of a forest meander system do not necessarily all have exterior arches. See example. Those that do are called tame (A060066). - Andrew Howroyd, Feb 02 2025

			An example of a 2 component forest meander system with 8 crossings that is not tame:
      ________
     / ______ \
    / /      \ \
   / / /\  /\ \ \
   \ \ \/ / /  \/
    \ \__/ /
     \____/

Roland Bacher, Meander algebras

Row sums of A380368.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

1 <= A060066(n) <= a(n) <= A060174(n) <= A060198(n) <= 16^n. - Andrew Howroyd, Feb 02 2025

More terms from Sascha Kurz, Mar 25 2002
a(15)-a(20) from Andrew Howroyd, Nov 22 2015
a(0)=1 prepended and a(21)-a(22) from Andrew Howroyd, Jan 31 2025

A060198 Dimensions of graded algebra associated with forest meanders.

Original entry on oeis.org

1, 16, 240, 3552, 52224, 764672, 11163936, 162631712, 2365037376, 34344187424, 498139336992, 7217820903328, 104490673015136, 1511512136496064, 21849735526096256, 315654728421607968, 4557598148470097472, 65771755517857808768, 948727279133187672224, 13679121303056997294080, 197153311343380929158240

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with 2n crossings which can be open on both sides and whose loops are all even. A loop is even if it has its least point at an even index with crossings numbered from zero. This definition excludes slices such as (open, open, close, close) since the inner loop is not even. It also excludes (up, open, close, down) even though the loop is not contained in another. - Andrew Howroyd, Feb 07 2025

			a(1) = 240 = 256 - 16 = 4^4 - 4^2 counts all length 4 sequences of open, close, up and down steps, excluding those that have an open + close pair as the central two elements. - _Andrew Howroyd_, Feb 07 2025

Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

Offset corrected and a(7) onwards from Andrew Howroyd, Feb 06 2025

A060111 Dimensions of graded algebra associated with meanders.

Original entry on oeis.org

1, 4, 15, 56, 207, 764, 2805, 10288, 37609, 137380, 500655, 1823440, 6629423, 24090332, 87418221, 317085352, 1148825185, 4160744164, 15054719697, 54454345624, 196805925995, 711077858188, 2567375653681, 9267176552040, 33430012251123, 120565130387572, 434578910451203

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

Number of meander slices with n crossings which can be open on both sides and contain no closed loops. These are called open meandric systems in the Bobier and Sawada reference. - Andrew Howroyd, Feb 07 2025

Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 0..27 from B. Bobier and J. Sawada)
Roland Bacher, Meander algebras, Institut Fourier, UMR 5582, Laboratoire de Mathématiques, 1999.
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages. [The final term for a(28) is incorrect].

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 26 2001

A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.

Original entry on oeis.org

1, 3, 2, 13, 16, 106, 166, 1073, 1934, 12142, 24076, 147090, 312906, 1865772, 4191822, 24463905, 57433950, 328887346, 800740450, 4508608610, 11319707546, 62781858592, 161841539812, 885513974674, 2335765140994, 12624162072740, 33979681977530, 181611275997040

Offset: 1

N. J. A. Sloane, Apr 10 2001

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Roland Bacher, Meander algebras

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

Cf. A018224.

seq(n) = Vec(1 - 1/sum(k=0, n, binomial(k, k\2)^2*x^k, O(x*x^n))) \\ Andrew Howroyd, Feb 07 2025

G.f.: 1 - 1/B(x) where B(x) is the g.f. of A018224. - Andrew Howroyd, Feb 07 2025

a(11) onwards from Andrew Howroyd, Feb 07 2025

A380368 Triangle read by rows: T(n,k) is the number of closed forest meander systems with 2n crossings and k components.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 8, 6, 1, 0, 42, 42, 12, 1, 0, 262, 320, 130, 20, 1, 0, 1828, 2618, 1360, 310, 30, 1, 0, 13820, 22582, 14196, 4270, 630, 42, 1, 0, 110954, 203006, 149024, 55524, 11060, 1148, 56, 1, 0, 933458, 1886004, 1577712, 698952, 175560, 25032, 1932, 72, 1

Offset: 0

Andrew Howroyd, Jan 31 2025

A forest meander system is a meander system that does not have any components which are entirely enclosed by another. An equivalent condition is that all components have their least point at an odd index (if points are numbered from 1). The greatest point will then be at an even index.

Exactly half of all meander systems with two components are forest meander systems. This is because when the meander's permutation is rotated one step at a time, one meander will be enclosed in the other on every second step.

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,     8,     6,     1;
  0,    42,    42,    12,    1;
  0,   262,   320,   130,   20,   1;
  0,  1828,  2618,  1360,  310,  30,  1;
  0, 13820, 22582, 14196, 4270, 630, 42, 1;
  ...
The T(3,2) = 6 forest meander systems are the following and their reflections.
       ______
      / ____ \                 ___
     / /    \ \               /   \
 .. / /. /\ .\ \ ..   and .. / / \ \ . /\ ..
    \/   \/   \/             \/   \/   \/
        (2)                     (4)
.
There are also 6 systems that are not forest meander systems:
      ____                    ______
     / __ \                  /      \
 .. / /  \ \ ..      and .. / /\  /\ \ ..
    \ \/\/ /                \ \/ /  \/
     \____/                  \__/
       (2)                     (4)

Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows 0..20)
Roland Bacher, Meander algebras.

Row sums are A060148.
Column k=1 is A005315.
Column k=2 is half of A006657.
Main diagonal is A000012.
Second diagonal is A002378.
Cf. A008828 (all meander systems), A060174, A060198.

cp's OEIS Frontend

A060206 Number of rotationally symmetric closed meanders of length 4n+2.

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A060066 Number of tame meanders with 2n crossings.

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A060089 Dimensions of graded algebra associated with meanders (subalgebra version).

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A060148 Number of closed forest meander systems with 2n crossings.

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A060198 Dimensions of graded algebra associated with forest meanders.

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A060111 Dimensions of graded algebra associated with meanders.

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A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.

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A380368 Triangle read by rows: T(n,k) is the number of closed forest meander systems with 2n crossings and k components.

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