A334062 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements.

1, 3, 1, 9, 12, 1, 27, 81, 31, 1, 81, 432, 390, 65, 1, 243, 2025, 3330, 1365, 120, 1, 729, 8748, 22815, 17415, 3909, 203, 1, 2187, 35721, 135513, 166320, 70938, 9730, 322, 1, 6561, 139968, 728028, 1312038, 911358, 242004, 21816, 486, 1, 19683, 531441, 3630420, 9032310, 9294264, 4067658, 722316, 45090, 705, 1

Offset

1,2

Comments

T(n,k) is also the number of non-crossing configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n, see [Young].

For the case of partitions of {1..3n} into sets of 3, see A091320.

For the case of partitions of {1..2n} into sets of 2, see A001263.

Links

Table of n, a(n) for n=1..55.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Formula

G.f.: G(t, z) satisfies z*G^4 - (1 + z - t*z)*G + 1 = 0.

Example

Triangle starts:
    1;
    3,    1;
    9,   12,    1;
   27,   81,   31,    1;
   81,  432,  390,   65,   1;
  243, 2025, 3330, 1365, 120, 1;
  ...
For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), (1,2,7,8),(3,4,5,6), and (1,2,3,8),(4,5,6,7); hence T(2,1) = 3.

Crossrefs

Row sums are A002293.

Column 2 is A069996.

Cf. A001263, A091320, A334063.

Sequence in context: A242499 A354622 A173020 * A157383 A232598 A174510

Adjacent sequences: A334059 A334060 A334061 * A334063 A334064 A334065

Keyword

nonn,tabl

Author

Donovan Young, May 28 2020

Status

approved