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

Author

Andrew Howroyd, Apr 04 2020

Keywords

• nonn
• tabl

Comments

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.

Examples

			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 ...
  ...
		

Links

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

Crossrefs

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.