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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..38.

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).

Formula

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

Example

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).

Crossrefs

Cf. A005315.

Sequence in context: A054993 A188912 A229285 * A005315 A182520 A121635

Adjacent sequences: A339457 A339458 A339459 * A339461 A339462 A339463

Keyword

tabf,nonn

Author

Gerasimos Pergaris, Dec 06 2020

Status

approved