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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.

%I A334060 #11 Jul 23 2025 16:01:33
%S A334060 1,0,1,7,3,0,219,56,5,0,12861,2352,183,4,0,1215794,174137,11145,323,1,
%T A334060 0,169509845,19970411,1078977,30833,334,0,0,32774737463,3280250014,
%U A334060 153076174,4056764,55379,206,0,0,8400108766161,730845033406,29989041076,727278456,10341101,67730,70,0,0
%N 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.
%C A334060 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 A334060 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334060 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 A334060 Triangle begins:
%e A334060       1;
%e A334060       0,    1;
%e A334060       7,    3,   0;
%e A334060     219,   56,   5, 0;
%e A334060   12861, 2352, 183, 4, 0;
%e A334060   ...
%e A334060 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 A334060 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 A334060 (PARI)
%o A334060 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 A334060 { my(A=T(8)); for(n=1, #A, print(A[n])) }
%Y A334060 Row sums are A025035.
%Y A334060 Column k=0 is column 0 of A334056.
%Y A334060 Cf. A079267, A334056, A334057, A334058, A334059, A325753.
%K A334060 nonn,tabl
%O A334060 0,4
%A A334060 _Donovan Young_, May 26 2020