A334061 Triangle read by rows: T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k disjoint strings of adjacent sets, each being a contiguous set of elements

1, 0, 1, 31, 4, 0, 5474, 292, 9, 0, 2554091, 72318, 1206, 10, 0, 2502018819, 43707943, 438987, 2871, 5, 0, 4456194509950, 52717010017, 351487598, 1622954, 4355, 1, 0, 13077453070386914, 111615599664989, 528618296314, 1764575884, 4080889, 4385, 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 4n, see [Young].

Links

Table of n, a(n) for n=0..35.

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

Formula

G.f.: Sum_{j>=0} (4*j)! * y^j * (1-(1-z)*y)^(4*j+1) / (j! * 24^j * (1-(1-z)*y^2)^(4*j+1)).

Example

Triangle begins:
        1;
        0,     1;
       31,     4,   0;
     5474,   292,   9,  0;
  2554091, 72318,1206, 10, 0;
  ...
For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), as well as (1,2,7,8),(3,4,5,6) and also (1,2,3,8),(4,5,6,7) (i.e. configurations with a single contiguous set) and (1,2,3,4),(5,6,7,8) (i.e. two adjacent contiguous sets); hence T(2,1) = 4.

Mathematica

CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(4*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]

Prog

(PARI)
T(n)={my(v=Vec(sum(j=0, n, (4*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(4*j+1) / (j! * 24^j * (1-(1-y)*x^2 + O(x*x^n))^(4*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }

Crossrefs

Row sums are A025036.

Column k=0 is column 0 of A334057.

Cf. A079267, A334056, A334057, A334058, A334059, A334060, A325753.

Sequence in context: A107811 A109837 A128372 * A040942 A153072 A040943

Adjacent sequences: A334058 A334059 A334060 * A334062 A334063 A334064

Keyword

nonn,tabl

Author

Donovan Young, May 26 2020

Status

approved