A005315 -id:A005315 - cp's OEIS frontend

A339460 Triangle read by rows: T(n,k) is the number of k-element equivalence classes of closed meanders with 2n points.

1, 2, 8, 42, 262, 1820, 4, 13756, 32, 110394, 280, 928790, 2328, 4, 8110104, 21294, 56, 73040142, 191396, 540, 24, 674775338, 1798624, 5214, 472, 6370633938, 17113152, 48240, 6482, 32, 61269105780, 168043112, 450616, 83804, 464, 32, 0, 4

Offset: 1

Gerasimos Pergaris, Dec 06 2020

Two closed meanders s and t with 2n points are equivalent iff their corresponding permutations s(1) s(2) ... s(2n) and t(1) t(2) ... t(2n) have the same absolute difference sequence, i.e. |s(i+1) - s(i)| = |t(i+1) - t(i)| for all i = 1,2,..,2n, where s(1) = t(1) = s(2n+1) = t(2n+1) = 1.

			Triangle begins:
            1;
            2;
            8;
           42;
          262;
         1820,         4;
        13756,        32;
       110394,       280;
       928790,      2328,      4;
      8110104,     21294,     56;
     73040142,    191396,    540,    24;
    674775338,   1798624,   5214,   472;
   6370633938,  17113152,  48240,  6482,  32;
  61269105780, 168043112, 450616, 83804, 464, 32, 0, 4;
  ...
For n = 6 there exist four 2-element equivalence classes:
1st class consists of permutations (1, 2, 5, 6, 7, 4, 3, 8, 9, 12, 11, 10) and (1, 2, 5, 4, 3, 6, 7, 12, 11, 8, 9, 10) having difference sequence: (1, 3, 1, 1, 3, 1, 5, 1, 3, 1, 1, 9).
2nd class consists of permutations (1, 12, 9, 10, 11, 8, 7, 2, 3, 6, 5, 4) and (1, 12, 9, 8, 7, 10, 11, 6, 5, 2, 3, 4) having difference sequence: (11, 3, 1, 1, 3, 1, 5, 1, 3, 1, 1, 3).
3rd class consists of permutations (1, 10, 9, 8, 11, 12, 7, 6, 3, 4, 5, 2) and (1, 10, 11, 12, 9, 8, 3, 4, 7, 6, 5, 2) having difference sequence: (9, 1, 1, 3, 1, 5, 1, 3, 1, 1, 3, 1).
4th class consists of permutations (1, 4, 5, 6, 3, 2, 7, 8, 11, 10, 9, 12) and (1, 4, 3, 2, 5, 6, 11, 10, 7, 8, 9, 12) having difference sequence: (3, 1, 1, 3, 1, 5, 1, 3, 1, 1, 3, 11).

M. De Biasi, Permutation Reconstruction from Differences, Electronic Journal of Combinatorics, Volume 21 No. 4 (2014), P4.3 (23 pages).
A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018).
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).

Cf. A005315.

Sum_{k >= 1} k*T(n,k) = A005315(n) (closed meandric numbers).

A363910 Triangular array read by rows: T(n,k) = the number of closed meanders with n top arches and k closed meanders in the reduction of the closed meander by the reverse of the exterior arch splitting algorithm.

Original entry on oeis.org

1, 0, 2, 0, 2, 6, 0, 6, 14, 22, 0, 28, 56, 86, 92, 0, 162, 298, 428, 518, 422, 0, 1076, 1868, 2562, 3096, 3144, 2074, 0, 7852, 13076, 17292, 20624, 21990, 19366, 10754

Offset: 1

Roger Ford, Jun 27 2023

The terms of this sequence can also be derived from sequences of consecutively numbered stamps folded with stamp 1 on top.

			n\k  1     2      3     4     5     6     7     8
1:   1
2:   0     2
3:   0     2      6
4:   0     6     14    22
5:   0    28     56    86    92
6:   0   162    298   428   518   422
7:   0  1076   1868  2562  3096  3144  2074
8:   0  7852  13076 17292 20624 21990 19366 10754
Closed meander:         Closed meander split with bottom rotated right
4 top arches            to form top of semi-meander with 8 arches
    ______                   ______
   / ____ \                 / ____ \
  / / __ \ \               / / __ \ \              __
 / / /  \ \ \             / / /  \ \ \            /  \
/ / / /\ \ \ \           / / / /\ \ \ \  /\  /\  / /\ \
\ \/ /  \/  \/           binary representation of semi-meander
 \__/                    1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0
                    Semi-meander top arches with no covering center arch  =  cm
                              START:          center |
Reduction of semi-meander:    1  1  1  1  0  0  0  0  1  0  1  0  1  1  0  0 cm(1)
Combine end of first arch     1  1  1  1  0  0  0  0e 1  0  1  0  1s 1  0  0
Oe with beginning of last        1  1  1  0  0  0  1  1  0  1  0  0  1  0
arch 1s.  0e...1s becomes        1  1  1  0  0  0e 1  1  0  1  0  0  1s 0
1...0 in the next line. The         1  1  0  0  1  1  1  0  1  0  0  0
starting 1 and ending 0             1  1  0  0e 1s 1  1  0  1  0  0  0
are removed in the next line           1  0  1  0  1  1  0  1  0  0
reducing number of arches.             1  0e 1  0  1s 1  0  1  0
by one.                                   1  1  0  0  1  0  1  0             cm(2)
                                          1  1  0  0e 1  0  1s 0
                                             1  0  1  1  0  0
                                             1  0e 1s 1  0  0
                                                1  0  1  0                   cm(3)
  Example: T(4,3) 4 starting top arches with 3 closed meanders in history.

Cf. A005315 (row sums), A001181, A005316, A000682.

T(n,n) = A001181(n).

T(n,2) = A005316(2*n-4)*2 for n > 1.

cp's OEIS Frontend

A339460 Triangle read by rows: T(n,k) is the number of k-element equivalence classes of closed meanders with 2n points.

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