A333734 -id:A333734 - 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.

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).

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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