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

Author

Donovan Young, May 28 2020

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

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

Crossrefs

Row sums are A002293.

Column 2 is A069996.

Cf. A001263, A091320, A334063.

Formula

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