A221961 -id:A221961 - cp's OEIS frontend

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

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

N. J. A. Sloane, Feb 23 2018

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].

			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, ...,
  ...

Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017. Also Discrete Math., 341 (2018), 341-349.

Cf. A299904, A299905.
See A299907 for main diagonal (i.e. square matrices).
See also A000629, A221961 for symmetric square matrices.
See A099594 for lonesum (0,1) matrices.

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 *)

More terms from Jean-François Alcover, Feb 24 2018

Name corrected by Alexander Karpov, Oct 19 2019

A299907 Number of decomposable lonesum n X n (0,1) matrices.

Original entry on oeis.org

1, 2, 16, 344, 13528, 833432, 73871416, 8893109864, 1394602938808, 275985896665592, 67227147723919096, 19756312865302785224, 6889612105281125091448, 2811911251972519547757272, 1327454489179890318710048056, 717555570945004782603934710824

Offset: 0

N. J. A. Sloane, Feb 23 2018

nonn

Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017. Also Discrete Math., 341 (2018), 341-349.

Main diagonal of A299906.
See also A000629, A221961 for symmetric square lonesum matrices.
See A099594 for lonesum (0,1) matrices.

a[n_] := Sum[Binomial[j - 1 , k - 1 ]*j!^2*StirlingS2[n + 1, j + 1]^2/k!, {k, 0, n}, {j, k, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 24 2018 *)

More terms from Jean-François Alcover, Feb 24 2018

cp's OEIS Frontend

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

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