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

0,4

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].

Links

Table of n, a(n) for n=0..44.

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

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)).

Example

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.

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}]

Prog

(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])) }

Crossrefs

Row sums are A025035.

Column k=0 is column 0 of A334056.

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

Sequence in context: A327574 A253905 A154159 * A378101 A083803 A370325

Adjacent sequences: A334057 A334058 A334059 * A334061 A334062 A334063

Keyword

nonn,tabl

Author

Donovan Young, May 26 2020

Status

approved