A333165 -id:A333165 - 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).

A333164 Number of 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero.

Original entry on oeis.org

1, 2, 18, 1760, 1944530, 26615510712, 4762109992158288, 11528251571501588791296, 386860001875783390762182911250, 183238648400953515891207813311894423000, 1242327573587456839123512395835593196519781229768, 121917941188016997420391711911475819481799846311888884541952

Offset: 0

Andrew Howroyd, Mar 11 2020

nonn

a(n) is the number of 2n X 2n symmetric binary matrices with n ones in every row and column.

			The a(1) = 2 matrices are:
  [ 1 -1]  [-1  1]
  [-1  1]  [ 1 -1]

Central coefficients of A333157.

Cf. A213793, A333165, A333166.

a(n) = A333157(2*n, n).

a(n) = A213793(2*n). - Andrew Howroyd, Apr 08 2020

A333166 Number of n-regular graphs on 2n unlabeled vertices with half-edges.

Original entry on oeis.org

1, 2, 3, 12, 118, 9638, 10622074, 135037240786, 18621890255342234, 28688490385422625653266, 511030957184968000138445253202

Offset: 0

Andrew Howroyd, Mar 12 2020

A half-edge is like a loop except it only adds 1 to the degree of its vertex.

a(n) is the number of non-isomorphic 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero where isomorphism is up to simultaneous permutation of rows and columns. The case where rows and columns can be permuted independently is covered by A333165.

			The a(1) = 1 matrix is:
  [+ -]
  [- +]
.
The a(2) = 2 matrices are:
  [+ + - -]   [- - + +]   [+ + - -]
  [+ + - -]   [- - + +]   [+ - + -]
  [- - + +]   [+ + - -]   [- + - +]
  [- - + +]   [+ + - -]   [- - + +]

Central coefficients of A333161.

a(n) = A333161(2*n, n).

Cf. A333164, A333165.

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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A333164 Number of 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero.

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A333166 Number of n-regular graphs on 2n unlabeled vertices with half-edges.

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