0,5

Also table read by rows: for 0 < k < n, a(n, k) = number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k vertices, up to isomorphism.

a(n, k) is the number of isomorphism classes of finite subdirectly irreducible almost distributive lattices in which the N-quotient has k upper covers and (n - k) lower covers. - David Wasserman (wasserma(AT)spawar.navy.mil), Feb 11 2002

Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

[1], [1, 1], [1, 3, 1], [1, 5, 5, 1], [1, 8, 17, 8, 1], ...; There are 17 bipartite graphs with 6 vertices, no isolated vertices and a distinguished bipartite block with 3 vertices, or equivalently, there are 17 3 X 3 binary matrices with no zero rows or columns, up to row and column permutation:

[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]

[0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 1] [0 1 1] [0 1 1] [1 1 0]

[1 1 0] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 0] [1 1 1] [1 1 0]

and

[0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]

[1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]

[1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].

Row sums give A055192. See A122083 for another version of this triangle.

Cf. A049312, A048194, A028657, A049311, A024206, A055609, A055082-A055084.

Sequence in context: A100936 A086620 A137897 this_sequence A125690 A108553 A158418

Adjacent sequences: A056149 A056150 A056151 this_sequence A056153 A056154 A056155

nonn,tabl

Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 29 2000