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A334062 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements.

%I A334062 #11 Apr 22 2022 10:39:15
%S A334062 1,3,1,9,12,1,27,81,31,1,81,432,390,65,1,243,2025,3330,1365,120,1,729,
%T A334062 8748,22815,17415,3909,203,1,2187,35721,135513,166320,70938,9730,322,
%U A334062 1,6561,139968,728028,1312038,911358,242004,21816,486,1,19683,531441,3630420,9032310,9294264,4067658,722316,45090,705,1
%N A334062 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements.
%C A334062 T(n,k) is also the number of non-crossing configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n, see [Young].
%C A334062 For the case of partitions of {1..3n} into sets of 3, see A091320.
%C A334062 For the case of partitions of {1..2n} into sets of 2, see A001263.
%H A334062 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334062 G.f.: G(t, z) satisfies z*G^4 - (1 + z - t*z)*G + 1 = 0.
%e A334062 Triangle starts:
%e A334062     1;
%e A334062     3,    1;
%e A334062     9,   12,    1;
%e A334062    27,   81,   31,    1;
%e A334062    81,  432,  390,   65,   1;
%e A334062   243, 2025, 3330, 1365, 120, 1;
%e A334062   ...
%e A334062 For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), (1,2,7,8),(3,4,5,6), and (1,2,3,8),(4,5,6,7); hence T(2,1) = 3.
%Y A334062 Row sums are A002293.
%Y A334062 Column 2 is A069996.
%Y A334062 Cf. A001263, A091320, A334063.
%K A334062 nonn,tabl
%O A334062 1,2
%A A334062 _Donovan Young_, May 28 2020