A333888 -id:A333888

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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
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

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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

Andrew Howroyd, Table of n, a(n) for n = 0..377

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

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.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 08 2020

STATUS

approved

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