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%I A184271 #48 Jul 05 2024 11:18:46
%S A184271 2,3,3,4,7,4,6,14,14,6,8,40,64,40,8,14,108,352,352,108,14,20,362,2192,
%T A184271 4156,2192,362,20,36,1182,14624,52488,52488,14624,1182,36,60,4150,
%U A184271 99880,699600,1342208,699600,99880,4150,60,108,14602,699252,9587580,35792568
%N A184271 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).
%C A184271 This is a 2-dimensional generalization of binary necklaces (A000031). A toroidal array or necklace can be defined either as an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns, or as a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns. - _Gus Wiseman_, Feb 04 2019
%H A184271 Alois P. Heinz, <a href="/A184271/b184271.txt">Antidiagonals n = 1..100, flattened</a> (first 95 terms from R. H. Hardin)
%H A184271 S. N. Ethier, <a href="http://arxiv.org/abs/1301.2352">Counting toroidal binary arrays</a>, arXiv preprint arXiv:1301.2352, 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">J. Int. Seq. 16 (2013) #13.4.7</a> .
%H A184271 S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html">J. Int. Seq. 18 (2015) # 15.8.3</a>.
%H A184271 Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.
%H A184271 Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv:2311.13072 [math.CO], 2023.
%H A184271 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pólya_enumeration_theorem">Pólya enumeration theorem</a>
%F A184271 T(n,k) = (1/(nk))*Sum_{ c divides n } Sum_{ d divides k } phi(c)*phi(d)*2^(nk/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - _Stewart N. Ethier_, Aug 24 2012
%e A184271       1     2        3           4            5             6              7
%e A184271 ----------------------------------------------------------------------------
%e A184271 1:    2     3        4           6            8            14             20
%e A184271 2:    3     7       14          40          108           362           1182
%e A184271 3:    4    14       64         352         2192         14624          99880
%e A184271 4:    6    40      352        4156        52488        699600        9587580
%e A184271 5:    8   108     2192       52488      1342208      35792568      981706832
%e A184271 6:   14   362    14624      699600     35792568    1908897152   104715443852
%e A184271 7:   20  1182    99880     9587580    981706832  104715443852 11488774559744
%e A184271 8:   36  4150   699252   134223976  27487816992 5864063066500
%e A184271 9:   60 14602  4971184  1908881900 781874936816
%e A184271 10: 108 52588 35792568 27487869472
%e A184271 From _Gus Wiseman_, Feb 04 2019: (Start)
%e A184271 Inequivalent representatives of the T(2,3) = 14 toroidal necklaces:
%e A184271   [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1]
%e A184271   [0 0 0] [0 0 1] [0 1 1] [1 1 1] [0 0 1] [0 1 0] [0 1 1]
%e A184271 .
%e A184271   [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
%e A184271   [1 0 1] [1 1 0] [1 1 1] [0 1 1] [1 0 1] [1 1 1] [1 1 1]
%e A184271 (End)
%t A184271 a[n_, k_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[k, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n k/LCM[c, d]), 0], {d, 1, k}], 0], {c, 1, n}]/(n k)
%t A184271 (* second program *)
%t A184271 neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
%t A184271 Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],neckmatQ]],{n,8},{k,n-1}] (* _Gus Wiseman_, Feb 04 2019 *)
%Y A184271 Columns 1-8 are A000031, A184264, A184265, A184266, A184267, A184268, A184269, A184270.
%Y A184271 Main diagonal is A179043.
%Y A184271 Cf. A001037 (binary Lyndon words), A008965, A323858, A323859 (binary toroidal necklaces of size n), A323861 (aperiodic version), A323865, A323870 (normal toroidal necklaces), A323872.
%K A184271 nonn,tabl
%O A184271 1,1
%A A184271 _R. H. Hardin_, Jan 10 2011

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