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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

%I #9 Apr 22 2022 10:39:09

%S 1,0,1,7,3,0,219,56,5,0,12861,2352,183,4,0,1215794,174137,11145,323,1,

%T 0,169509845,19970411,1078977,30833,334,0,0,32774737463,3280250014,

%U 153076174,4056764,55379,206,0,0,8400108766161,730845033406,29989041076,727278456,10341101,67730,70,0,0

%N 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

%C 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].

%H Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.

%F 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)).

%e Triangle begins:

%e 1;

%e 0, 1;

%e 7, 3, 0;

%e 219, 56, 5, 0;

%e 12861, 2352, 183, 4, 0;

%e ...

%e 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.

%t 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}]

%o (PARI)

%o 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))}

%o { my(A=T(8)); for(n=1, #A, print(A[n])) }

%Y Row sums are A025035.

%Y Column k=0 is column 0 of A334056.

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

%K nonn,tabl

%O 0,4

%A _Donovan Young_, May 26 2020