A333159 Triangle read by rows: T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column up to permutation of rows and columns.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 4, 12, 12, 4, 1, 1, 1, 1, 7, 31, 66, 31, 7, 1, 1, 1, 1, 8, 90, 433, 433, 90, 8, 1, 1, 1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1, 1, 1, 14, 938, 30404, 171984, 171984, 30404, 938, 14, 1, 1

Offset

0,13

Comments

Rows and columns may be permuted independently. The case that rows and columns must be permuted together is covered by A333161.

T(n,k) is the number of k-regular bicolored graphs on 2n unlabeled nodes which are invariant when the two color classes are exchanged.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..230

Formula

T(n,k) = T(n,n-k).

Example

Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,   1;
  1, 1,  2,   1,    1;
  1, 1,  2,   2,    1,    1;
  1, 1,  4,   5,    4,    1,    1;
  1, 1,  4,  12,   12,    4,    1,   1;
  1, 1,  7,  31,   66,   31,    7,   1,  1;
  1, 1,  8,  90,  433,  433,   90,   8,  1, 1;
  1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1;
  ...
The T(2,1) = 1 matrix is:
  [1 0]
  [0 1]
.
The T(4,2)= 2 matrices are:
  [1 1 0 0]   [1 1 0 0]
  [1 1 0 0]   [1 0 1 0]
  [0 0 1 1]   [0 1 0 1]
  [0 0 1 1]   [0 0 1 1]

Crossrefs

Columns k=0..4 are A000012, A000012, A002865, A000840, A000843.

Row sums are A333160.

Central coefficients are A333165.

Cf. A008327, A122082, A333157, A133687, A333161.

Sequence in context: A047996 A227690 A063686 * A008327 A133687 A215870

Adjacent sequences: A333156 A333157 A333158 * A333160 A333161 A333162

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 10 2020

Status

approved