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A333737
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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.
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10
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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
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OFFSET
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0,13
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COMMENTS
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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.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..377
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EXAMPLE
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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]
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CROSSREFS
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Rows n=0..5 are A000012, A000012, A008619, A106607, A333886, A333887.
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).
Cf. A318805, A333893.
Sequence in context: A330461 A332649 A321724 * A333733 A303929 A303694
Adjacent sequences: A333734 A333735 A333736 * A333738 A333739 A333740
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KEYWORD
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nonn,tabl
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AUTHOR
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Andrew Howroyd, Apr 08 2020
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STATUS
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approved
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