A000843 -id:A000843 - cp's OEIS frontend

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

Andrew Howroyd, Mar 10 2020

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.

			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]

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

Columns k=0..4 are A000012, A000012, A002865, A000840, A000843.
Row sums are A333160.
Central coefficients are A333165.
Cf. A008327, A122082, A333157, A133687, A333161.

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

A000840 Number of cubic bicolored graphs on n unlabeled nodes admitting an automorphism exchanging the colors.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 12, 31, 90, 285, 938, 3285, 11983, 45390, 177803, 718390, 2986407, 12749364, 55802982, 250068732, 1145923828, 5363795830, 25620207380, 124767647097, 618983876918, 3126035142910, 16060182735947, 83883575376862, 445164927249466, 2399098651337048

Offset: 1

Brendan McKay and Eric Rogoyski

nonn

Column k=3 of A333159.

Cf. A000512, A000843, A004066, A008325.

a(11)-a(30) from Andrew Howroyd, Mar 10 2020

cp's OEIS Frontend

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.

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A000840 Number of cubic bicolored graphs on n unlabeled nodes admitting an automorphism exchanging the colors.

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