A377060 -id:A377060 - cp's OEIS frontend

A333733 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to k up to permutations of rows and columns.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 9, 12, 7, 1, 1, 1, 1, 4, 13, 43, 31, 11, 1, 1, 1, 1, 4, 22, 106, 264, 103, 15, 1, 1, 1, 1, 5, 30, 321, 1856, 2804, 383, 22, 1, 1, 1, 1, 5, 45, 787, 12703, 65481, 44524, 1731, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 04 2020

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 A257493. Burnside's lemma can be used to extend this method to the unlabeled case.

			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  3   5     9      13        22         30 ...
  4 | 1 1  5  12    43     106       321        787 ...
  5 | 1 1  7  31   264    1856     12703      71457 ...
  6 | 1 1 11 103  2804   65481   1217727   16925049 ...
  7 | 1 1 15 383 44524 3925518 224549073 8597641912 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)

Rows n=0..5 are A000012, A000012, A008619, A052282, A052280, A333735.
Columns k=0..5 are A000012, A000012, A000041, A232215, A232216, A333736.
Main diagonal is A333734.
Cf. A133687, A167625, A257493, A333330, A377060.

A333734 Number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to n up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 2, 5, 43, 1856, 1217727, 8597641912, 646296747486387, 535435113671568180963, 5081029530811947425598907884, 570680215340337514993573217774604779, 779646755088025699677478853259568262608053838

Offset: 0

Andrew Howroyd, Apr 04 2020

			The a(2) = 2 matrices are:
  [1 1]  [2 0]
  [1 1]  [0 2]
.
The a(3) = 5 matrices are:
  [1 1 1]   [2 1 0]   [2 1 0]   [3 0 0]   [3 0 0]
  [1 1 1]   [1 1 1]   [0 2 1]   [0 2 1]   [0 3 0]
  [1 1 1]   [0 1 2]   [1 0 2]   [0 1 2]   [0 0 3]

Main diagonal of A333733 and A377060.

Cf. A110058.

a(11)-a(12) from Andrew Howroyd, Oct 14 2024

A377061 Number of inequivalent n X 3 nonnegative integer arrays with row sums 3 and column sums n up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 2, 5, 9, 14, 28, 44, 73, 118, 184, 276, 422, 610, 880, 1248, 1742, 2383, 3250, 4341, 5758, 7550, 9808, 12600, 16103, 20352, 25572, 31890, 39519, 48620, 59548, 72432, 87703, 105629, 126642, 151077, 179572, 212428, 250408, 294024, 344052, 401106, 466260, 540110, 623908

Offset: 0

Andrew Howroyd, Oct 14 2024

nonn

			The a(2) = 2 matrices are:
  [2 1 0]   [1 1 1]
  [0 1 2]   [1 1 1]
.
The a(3) = 5 matrices are:
  [3 0 0]   [3 0 0]   [2 1 0]   [2 1 0]   [1 1 1]
  [0 3 0]   [0 2 1]   [1 1 1]   [1 0 2]   [1 1 1]
  [0 0 3]   [0 1 2]   [0 1 2]   [0 2 1]   [1 1 1]

Column k=3 of A377060.

Cf. A172743, A377062.

A377062 Number of inequivalent n X 4 nonnegative integer arrays with row sums 4 and column sums n up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 3, 9, 43, 147, 661, 2649, 10738, 40427, 146206, 498705, 1625621, 5049556, 15044485, 43044068, 118731621, 316347549, 816409799, 2044571309, 4979112445, 11810405775, 27331117859, 61792085482, 136670650003, 296071435108, 628900592847, 1311195696010, 2685731232674, 5409289235300, 10721363532282

Offset: 0

Andrew Howroyd, Oct 14 2024

nonn

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

Column k=4 of A377060.

Cf. A172816, A377061.

cp's OEIS Frontend

A333733 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to k up to permutations of rows and columns.

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A333734 Number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to n up to permutations of rows and columns.

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A377061 Number of inequivalent n X 3 nonnegative integer arrays with row sums 3 and column sums n up to permutations of rows and columns.

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A377062 Number of inequivalent n X 4 nonnegative integer arrays with row sums 4 and column sums n up to permutations of rows and columns.

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