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A334063 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..5n} into n sets of 5 with k of the sets being a contiguous set of elements.

%I A334063 #12 Apr 22 2022 10:39:19
%S A334063 1,4,1,16,18,1,64,168,52,1,256,1216,936,121,1,1024,7680,11040,3760,
%T A334063 246,1,4096,44544,103040,67480,12264,455,1,16384,243712,827904,888160,
%U A334063 318976,34524,784,1,65536,1277952,5992448,9554944,5716704,1254512,86980,1278,1
%N A334063 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..5n} into n sets of 5 with k of the sets being a contiguous set of elements.
%C A334063 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 5n, see [Young].
%C A334063 For the case of partitions of {1..4n} into sets of 4, see A334062.
%C A334063 For the case of partitions of {1..3n} into sets of 3, see A091320.
%C A334063 For the case of partitions of {1..2n} into sets of 2, see A001263.
%H A334063 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334063 G.f.: G(t, z) satisfies z*G^5 - (1 + z - t*z)*G + 1 = 0.
%e A334063 Triangle starts:
%e A334063      1;
%e A334063      4,    1;
%e A334063     16,   18,     1;
%e A334063     64,  168,    52,    1;
%e A334063    256, 1216,   936,  121,   1;
%e A334063   1024, 7680, 11040, 3760, 246,  1;
%e A334063   ...
%e A334063 For n = 2 and k = 1 the configurations are (1,7,8,9,10), (2,3,4,5,6), (1,2,8,9,10),(3,4,5,6,7), (1,2,3,9,10), (4,5,6,7,8) and (1,2,3,4,10), (5,6,7,8,9); hence T(2,1) = 4.
%Y A334063 Row sums are A002294.
%Y A334063 Cf. A001263, A091320, A334062.
%K A334063 nonn,tabl
%O A334063 1,2
%A A334063 _Donovan Young_, May 28 2020