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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.

%I A334061 #11 Jul 23 2025 16:01:39
%S A334061 1,0,1,31,4,0,5474,292,9,0,2554091,72318,1206,10,0,2502018819,
%T A334061 43707943,438987,2871,5,0,4456194509950,52717010017,351487598,1622954,
%U A334061 4355,1,0,13077453070386914,111615599664989,528618296314,1764575884,4080889,4385,0,0
%N 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.
%C A334061 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].
%H A334061 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334061 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)).
%e A334061 Triangle begins:
%e A334061         1;
%e A334061         0,     1;
%e A334061        31,     4,   0;
%e A334061      5474,   292,   9,  0;
%e A334061   2554091, 72318,1206, 10, 0;
%e A334061   ...
%e A334061 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.
%t A334061 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}]
%o A334061 (PARI)
%o A334061 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))}
%o A334061 { my(A=T(8)); for(n=1, #A, print(A[n])) }
%Y A334061 Row sums are A025036.
%Y A334061 Column k=0 is column 0 of A334057.
%Y A334061 Cf. A079267, A334056, A334057, A334058, A334059, A334060, A325753.
%K A334061 nonn,tabl
%O A334061 0,4
%A A334061 _Donovan Young_, May 26 2020