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A333737
Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.
10
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, 33, 29, 11, 1, 1, 1, 1, 4, 20, 74, 142, 79, 15, 1, 1, 1, 1, 5, 28, 163, 556, 742, 225, 22, 1, 1, 1, 1, 5, 39, 319, 1919, 5369, 4454, 677, 30, 1, 1
OFFSET
0,13
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 A188403. Burnside's lemma as applied in A318805 can be used to extend this method to the unlabeled case.
LINKS
EXAMPLE
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 20 28 ...
4 | 1 1 5 12 33 74 163 319 ...
5 | 1 1 7 29 142 556 1919 5793 ...
6 | 1 1 11 79 742 5369 31781 156191 ...
7 | 1 1 15 225 4454 64000 692599 5882230 ...
...
The T(3,3) = 5 matrices are:
[0 0 3] [0 1 2] [0 1 2] [1 0 2] [1 1 1]
[0 3 0] [1 1 1] [1 2 0] [0 3 0] [1 1 1]
[3 0 0] [2 1 0] [2 0 1] [2 0 1] [1 1 1]
CROSSREFS
Columns n=0..5 are A000012, A000012, A000041, A333888, A333889, A333890.
Main diagonal is A333738.
Cf. A188403 (labeled case), A333159 (binary), A333733 (not necessarily symmetric).
Sequence in context: A330461 A332649 A321724 * A333733 A303929 A303694
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 08 2020
STATUS
approved