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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.
5

%I #12 Oct 15 2024 00:02:04

%S 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,

%T 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,

%U 661,44,5,1,1,1,1,5,73,2649,25888,25888,2649,73,5,1,1

%N 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.

%C 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.

%H Andrew Howroyd, <a href="/A377060/b377060.txt">Table of n, a(n) for n = 0..324</a> (first 25 antidiagonals)

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

%e Array begins:

%e ==================================================

%e n\k | 0 1 2 3 4 5 6 7 ...

%e ----+---------------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 ...

%e 1 | 1 1 1 1 1 1 1 1 ...

%e 2 | 1 1 2 2 3 3 4 4 ...

%e 3 | 1 1 2 5 9 14 28 44 ...

%e 4 | 1 1 3 9 43 147 661 2649 ...

%e 5 | 1 1 3 14 147 1856 25888 346691 ...

%e 6 | 1 1 4 28 661 25888 1217727 55138002 ...

%e 7 | 1 1 4 44 2649 346691 55138002 8597641912 ...

%e ...

%Y Main diagonal is A333734.

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

%Y Cf. A333733, A333901, A377007.

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, Oct 14 2024