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A339460 Triangle read by rows: T(n,k) is the number of k-element equivalence classes of closed meanders with 2n points. 0
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
tabf,nonn
AUTHOR
Gerasimos Pergaris, Dec 06 2020
STATUS
approved

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Last modified July 7 06:35 EDT 2024. Contains 374063 sequences. (Running on oeis4.)