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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 (list; table; graph; refs; listen; history; text; internal format)
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.

Sequence in context: A330461 A332649 A321724 * A333733 A303929 A303694

Adjacent sequences:  A333734 A333735 A333736 * A333738 A333739 A333740

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 08 2020

STATUS

approved

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Last modified May 19 17:07 EDT 2022. Contains 353847 sequences. (Running on oeis4.)