A333162 -id:A333162 - cp's OEIS frontend

A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 8, 12, 8, 4, 1, 1, 4, 10, 24, 24, 10, 4, 1, 1, 5, 17, 70, 118, 70, 17, 5, 1, 1, 5, 24, 172, 634, 634, 172, 24, 5, 1, 1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1, 1, 6, 50, 1530, 35500, 187990, 187990, 35500, 1530, 50, 6, 1

Offset: 0

Andrew Howroyd, Mar 11 2020

A half-edge is like a loop except it only adds 1 to the degree of its vertex.
T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333159.
T(n,k) is the number of simple graphs on n unlabeled vertices with every vertex degree being either k or k-1.

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  2,   1;
  1, 3,  3,   3,    1;
  1, 3,  4,   4,    3,    1;
  1, 4,  8,  12,    8,    4,    1;
  1, 4, 10,  24,   24,   10,    4,   1;
  1, 5, 17,  70,  118,   70,   17,   5,  1;
  1, 5, 24, 172,  634,  634,  172,  24,  5, 1;
  1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1;
  ...
The a(2,1) = 2 adjacency matrices are:
  [0 1]  [1 0]
  [1 0]  [0 1]
.
The A(4,2) = 3 adjacency matrices are:
  [0 0 1 1]   [1 1 0 0]   [1 1 0 0]
  [0 0 1 1]   [1 1 0 0]   [1 0 1 0]
  [1 1 0 0]   [0 0 1 1]   [0 1 0 1]
  [1 1 0 0]   [0 0 1 1]   [0 0 1 1]

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

Columns k=0..3 are A000012, A004526(n+2), A186417, A333163.
Row sums are A333162.
Central coefficients are A333166.
Cf. A051031, A333157, A333159.

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

A333160 Number of non-isomorphic n X n symmetric binary matrices with an equal number of ones in every row and column up to permutation of rows and columns.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 17, 36, 146, 1066, 15419, 406684, 19284912, 1635874946, 249424764407, 68725494158824, 34418706513939926, 31487353344361957012, 52887877379630894268187, 163777247316556715401451972, 939121048579630147375554814224

Offset: 0

Andrew Howroyd, Mar 10 2020

nonn

a(n) is the number of regular bicolored graphs on 2n unlabeled nodes which are invariant when the two color classes are interchanged.

			The a(2) = 3 matrices are:
  [0 0]  [1 0]  [1 1]
  [0 0]  [0 1]  [1 1]

Row sums of A333159.

Cf. A122082, A322698, A333162.

cp's OEIS Frontend

A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

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