A377007 Array read by antidiagonals: T(n,k) is the number of inequivalent 2*n X 2*k binary matrices with all row sums k and column sums n up to permutations of rows and columns.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 7, 4, 1, 1, 1, 1, 5, 19, 19, 5, 1, 1, 1, 1, 7, 46, 194, 46, 7, 1, 1, 1, 1, 8, 132, 3144, 3144, 132, 8, 1, 1, 1, 1, 10, 345, 65548, 601055, 65548, 345, 10, 1, 1, 1, 1, 12, 951, 1272696, 128665248, 128665248, 1272696, 951, 12, 1, 1

Offset: 0

Author

Andrew Howroyd, Oct 12 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 A376935. 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   3       4           5               7                    8 ...
  3 | 1 1 3   7      19          46             132                  345 ...
  4 | 1 1 4  19     194        3144           65548              1272696 ...
  5 | 1 1 5  46    3144      601055       128665248          24124134235 ...
  6 | 1 1 7 132   65548   128665248    294494683312      607662931576945 ...
  7 | 1 1 8 345 1272696 24124134235 607662931576945 14584161564179926207 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..119 (first 15 antidiagonals)

Crossrefs

Main diagonal is A333740.

Columns k=0..6 are A000012, A000012, A001399, A377003, A377004, A377005, A377006.

Cf. A133687, A376935.

Formula

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