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A299906 Array read by antidiagonals: T(n,k) = number of n X k lonesum decomposable (0,1) matrices.

%I A299906 #24 Oct 26 2019 01:30:19
%S A299906 1,1,1,1,2,1,1,4,4,1,1,8,16,8,1,1,16,58,58,16,1,1,32,196,344,196,32,1,
%T A299906 1,64,634,1786,1786,634,64,1,1,128,1996,8528,13528,8528,1996,128,1,1,
%U A299906 256,6178,38578,90946,90946,38578,6178,256,1,1,512,18916,168344,564376,833432,564376,168344,18916,512,1
%N A299906 Array read by antidiagonals: T(n,k) = number of n X k lonesum decomposable (0,1) matrices.
%C A299906 A (0,1) n X k matrix is lonesum if the matrix is uniquely determined by its row-sum and column-sum vectors, that is, by the sum of its rows and the sum of its columns. For example, the 2 X 3 matrix [1,1,1 / 0,1,0] is the only matrix with column-sum vector [1,2,1] and row-sum vector [3,1].
%H A299906 Ken Kamano, <a href="https://arxiv.org/abs/1701.07157">Lonesum decomposable matrices</a>, arXiv:1701.07157 [math.CO], 2017. Also Discrete Math., 341 (2018), 341-349.
%e A299906 Array begins:
%e A299906   1,  1,   1,    1,     1,      1, ...,
%e A299906   1,  2,   4,    8,    16,     32, ...,
%e A299906   1,  4,  16,   58,   196,    634, ...,
%e A299906   1,  8,  58,  344,  1786,   8528, ...,
%e A299906   1, 16, 196, 1786, 13528,  90946, ...,
%e A299906   1, 32, 634, 8528, 90446, 833432, ...,
%e A299906   ...
%t A299906 T[n_, k_] := Sum[(Binomial[j-1, k0-1] * j!^2 * StirlingS2[k+1, j+1] * StirlingS2[n+1, j+1])/k0!, {k0, 0, k}, {j, k0, Min[k, n]}]; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 24 2018 *)
%Y A299906 Cf. A299904, A299905.
%Y A299906 See A299907 for main diagonal (i.e. square matrices).
%Y A299906 See also A000629, A221961 for symmetric square matrices.
%Y A299906 See A099594 for lonesum (0,1) matrices.
%K A299906 nonn,tabl
%O A299906 0,5
%A A299906 _N. J. A. Sloane_, Feb 23 2018
%E A299906 More terms from _Jean-François Alcover_, Feb 24 2018
%E A299906 Name corrected by _Alexander Karpov_, Oct 19 2019