A334060 Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements.

1, 0, 1, 7, 3, 0, 219, 56, 5, 0, 12861, 2352, 183, 4, 0, 1215794, 174137, 11145, 323, 1, 0, 169509845, 19970411, 1078977, 30833, 334, 0, 0, 32774737463, 3280250014, 153076174, 4056764, 55379, 206, 0, 0, 8400108766161, 730845033406, 29989041076, 727278456, 10341101, 67730, 70, 0, 0

Offset: 0

Author

Donovan Young, May 26 2020

Keywords

• nonn
• tabl

Comments

Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 3n, see [Young].

Examples

			Triangle begins:
      1;
      0,    1;
      7,    3,   0;
    219,   56,   5, 0;
  12861, 2352, 183, 4, 0;
  ...
For n=2 and k=1 the configurations are (1,5,6),(2,3,4) and (1,2,6),(3,4,5) (i.e. configurations with a single contiguous set) and (1,2,3),(4,5,6) (i.e. two adjacent contiguous sets); hence T(2,1) = 3.
		

Links

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

Crossrefs

Row sums are A025035.

Column k=0 is column 0 of A334056.

Cf. A079267, A334056, A334057, A334058, A334059, A325753.

Programs

Mathematica

CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]

PARI

T(n)={my(v=Vec(sum(j=0, n, (3*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(3*j+1) / (j! * 6^j * (1-(1-y)*x^2 + O(x*x^n))^(3*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }

Formula

G.f.: Sum_{j>=0} (3*j)! * y^j * (1-(1-z)*y)^(3*j+1) / (j! * 6^j * (1-(1-z)*y^2)^(3*j+1)).