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.

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

Author

Andrew Howroyd, Oct 14 2024

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 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.

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 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 ...
  ...
		

Links

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

Crossrefs

Main diagonal is A333734.

Columns k=0..4 are A000012, A000012, A008619, A377061, A377062.

Cf. A333733, A333901, A377007.

Formula

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