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.

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

Author

Andrew Howroyd, Apr 08 2020

Keywords

• nonn
• tabl

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.

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..377
• Ming Yean Lim, The number of irreducibles in the plethysm s_lambda[s_m], arXiv:2503.21108 [math.CO], 2025. See p. 8.

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.