A377007 -id:A377007 - cp's OEIS frontend

A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 20, 90, 20, 1, 1, 70, 1860, 1860, 70, 1, 1, 252, 44730, 297200, 44730, 252, 1, 1, 924, 1172556, 60871300, 60871300, 1172556, 924, 1, 1, 3432, 32496156, 14367744720, 116963796250, 14367744720, 32496156, 3432, 1, 1, 12870, 936369720, 3718394156400, 273957842462220, 273957842462220, 3718394156400, 936369720, 12870, 1

Offset: 0

Andrew Howroyd, Oct 11 2024

T(n,k) is the number of 2*n X 2*k {-1,1} matrices with all rows and columns summing to zero.

			Array begins:
========================================================================
n\k | 0   1       2           3               4                   5 ...
----+------------------------------------------------------------------
  0 | 1   1       1           1               1                   1 ...
  1 | 1   2       6          20              70                 252 ...
  2 | 1   6      90        1860           44730             1172556 ...
  3 | 1  20    1860      297200        60871300         14367744720 ...
  4 | 1  70   44730    60871300    116963796250     273957842462220 ...
  5 | 1 252 1172556 14367744720 273957842462220 6736218287430460752 ...
  ...

Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Main diagonal is A058527.
Columns 0..9 are A000012, A000984, A002896, A172556, A172555, A172557, A172558, A172559, A172560, A172554.
Cf. A008300, A195644, A333901, A334549, A377007 (up to permutations of rows and columns).

T(n, k)={
  local(M=Map(Mat([2*k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
  for(r=1, 2*n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
}

T(n,k) = T(k,n).

A377060 Array read by antidiagonals: T(n,k) is the number of inequivalent n X k nonnegative integer matrices with all column sums n and row sums k up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 3, 9, 9, 3, 1, 1, 1, 1, 4, 14, 43, 14, 4, 1, 1, 1, 1, 4, 28, 147, 147, 28, 4, 1, 1, 1, 1, 5, 44, 661, 1856, 661, 44, 5, 1, 1, 1, 1, 5, 73, 2649, 25888, 25888, 2649, 73, 5, 1, 1

Offset: 0

Andrew Howroyd, Oct 14 2024

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A333901. Burnside's lemma can be used to extend this method to the unlabeled case. This seems to require looping over partitions for both rows and columns.

			Array begins:
==================================================
n\k | 0 1 2  3    4      5        6          7 ...
----+---------------------------------------------
  0 | 1 1 1  1    1      1        1          1 ...
  1 | 1 1 1  1    1      1        1          1 ...
  2 | 1 1 2  2    3      3        4          4 ...
  3 | 1 1 2  5    9     14       28         44 ...
  4 | 1 1 3  9   43    147      661       2649 ...
  5 | 1 1 3 14  147   1856    25888     346691 ...
  6 | 1 1 4 28  661  25888  1217727   55138002 ...
  7 | 1 1 4 44 2649 346691 55138002 8597641912 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..324 (first 25 antidiagonals)

Main diagonal is A333734.
Columns k=0..4 are A000012, A000012, A008619, A377061, A377062.
Cf. A333733, A333901, A377007.

T(n,k) = T(k,n).

A377003 Number of inequivalent 2*n X 6 binary arrays with row sums 3 and column sums n up to row and column permutations.

Original entry on oeis.org

1, 1, 3, 7, 19, 46, 132, 345, 951, 2457, 6198, 14816, 34126, 75057, 159176, 325021, 642473, 1230711, 2292826, 4160165, 7370620, 12768904, 21671722, 36080204, 59009291, 94910598, 150300578, 234560237, 361082376, 548722631, 823817750

Offset: 0

Andrew Howroyd, Oct 12 2024

			The a(2) = 3 inequivalent matrices are:
  [ 1 1 1 0 0 0 ]    [ 1 1 1 0 0 0 ]    [ 1 1 1 0 0 0 ]
  [ 1 1 1 0 0 0 ]    [ 1 1 0 1 0 0 ]    [ 1 0 0 1 1 0 ]
  [ 0 0 0 1 1 1 ]    [ 0 0 1 0 1 1 ]    [ 0 1 0 1 0 1 ]
  [ 0 0 0 1 1 1 ]    [ 0 0 0 1 1 1 ]    [ 0 0 1 0 1 1 ]

Column k=3 of A377007.

Cf. A172556.

A377004 Number of inequivalent 2*n X 8 binary arrays with row sums 4 and column sums n up to row and column permutations.

Original entry on oeis.org

1, 1, 4, 19, 194, 3144, 65548, 1272696, 21855504, 328629713, 4362818853, 51658865022, 551021529952, 5342121192394, 47446872310092, 388746259849648, 2956322263908194, 20980847841062062, 139630332607991630, 875184633222730558, 5186438905325403553

Offset: 0

Andrew Howroyd, Oct 12 2024

nonn

			The a(2) = 4 inequivalent matrices are:
  [1 1 1 1 0 0 0 0]  [1 1 1 1 0 0 0 0 ]  [1 1 1 1 0 0 0 0]  [1 1 1 1 0 0 0 0]
  [1 1 1 1 0 0 0 0]  [1 1 1 0 1 0 0 0 ]  [1 1 0 0 1 1 0 0]  [1 1 0 0 1 1 0 0]
  [0 0 0 0 1 1 1 1]  [0 0 0 1 0 1 1 1 ]  [0 0 1 1 0 0 1 1]  [0 0 1 0 1 0 1 1]
  [0 0 0 0 1 1 1 1]  [0 0 0 0 1 1 1 1 ]  [0 0 0 0 1 1 1 1]  [0 0 0 1 0 1 1 1]

Column k=4 of A377007.

Cf. A172555, A377003.

A377005 Number of inequivalent 2*n X 10 binary arrays with row sums 5 and column sums n up to row and column permutations.

Original entry on oeis.org

1, 1, 5, 46, 3144, 601055, 128665248, 24124134235, 3846063693791, 523474723784499, 61451727919769422, 6288576962481252097, 566522424663964802533, 45327061007633462729853, 3246317954956652960733055, 209590042445523033214523199, 12275325090242910474009076242

Offset: 0

Andrew Howroyd, Oct 12 2024

nonn

Column k=5 of A377007.

Cf. A172557.

A377006 Number of inequivalent 2*n X 12 binary arrays with row sums 6 and column sums n up to row and column permutations.

Original entry on oeis.org

1, 1, 7, 132, 65548, 128665248, 294494683312, 607662931576945, 1078571634893111039, 1641745848357513999450, 2157522985252158679423406, 2468770667797731319051018701, 2480199767193520485788791333065

Offset: 0

Andrew Howroyd, Oct 12 2024

Column k=6 of A377007.

Cf. A172558.

cp's OEIS Frontend

A376935 Array read by antidiagonals: T(n,k) is the number of 2*n X 2*k binary matrices with all row sums k and column sums n.

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A377060 Array read by antidiagonals: T(n,k) is the number of inequivalent n X k nonnegative integer matrices with all column sums n and row sums k up to permutation of rows and columns.

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A377003 Number of inequivalent 2*n X 6 binary arrays with row sums 3 and column sums n up to row and column permutations.

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A377004 Number of inequivalent 2*n X 8 binary arrays with row sums 4 and column sums n up to row and column permutations.

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A377005 Number of inequivalent 2*n X 10 binary arrays with row sums 5 and column sums n up to row and column permutations.

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A377006 Number of inequivalent 2*n X 12 binary arrays with row sums 6 and column sums n up to row and column permutations.

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A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.