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A299906
Array read by antidiagonals: T(n,k) = number of n X k lonesum decomposable (0,1) matrices.
4
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, 1, 64, 634, 1786, 1786, 634, 64, 1, 1, 128, 1996, 8528, 13528, 8528, 1996, 128, 1, 1, 256, 6178, 38578, 90946, 90946, 38578, 6178, 256, 1, 1, 512, 18916, 168344, 564376, 833432, 564376, 168344, 18916, 512, 1
OFFSET
0,5
COMMENTS
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].
LINKS
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017. Also Discrete Math., 341 (2018), 341-349.
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, ...,
1, 2, 4, 8, 16, 32, ...,
1, 4, 16, 58, 196, 634, ...,
1, 8, 58, 344, 1786, 8528, ...,
1, 16, 196, 1786, 13528, 90946, ...,
1, 32, 634, 8528, 90446, 833432, ...,
...
MATHEMATICA
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 *)
CROSSREFS
See A299907 for main diagonal (i.e. square matrices).
See also A000629, A221961 for symmetric square matrices.
See A099594 for lonesum (0,1) matrices.
Sequence in context: A255256 A328887 A372067 * A117401 A144324 A331406
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Feb 23 2018
EXTENSIONS
More terms from Jean-François Alcover, Feb 24 2018
Name corrected by Alexander Karpov, Oct 19 2019
STATUS
approved