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A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

%I A056152 #13 Mar 26 2020 19:43:56
%S A056152 1,1,1,1,3,1,1,5,5,1,1,8,17,8,1,1,11,42,42,11,1,1,15,91,179,91,15,1,1,
%T A056152 19,180,633,633,180,19,1,1,24,328,2001,3835,2001,328,24,1,1,29,565,
%U A056152 5745,20755,20755,5745,565,29,1,1,35,930,15274,102089,200082,102089
%N A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.
%C A056152 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.
%C A056152 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_, Feb 11 2002
%C A056152 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.
%D A056152 J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.
%D A056152 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%H A056152 F. Harary, L. March and R. W. Robinson, <a href="https://doi.org/10.1068/b050031">On enumerating certain design problems in terms of bicolored graphs with no isolates</a>, Environment and Planning, B 5 (1978), 31-43. See Table 2.
%H A056152 F. Harary, L. March and R. W. Robinson, <a href="/A007139/a007139.pdf">On enumerating certain design problems in terms of bicolored graphs with no isolates</a>, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 2.
%e A056152 Triangle begins:
%e A056152   1;
%e A056152   1,  1;
%e A056152   1,  3,   1;
%e A056152   1,  5,   5,   1;
%e A056152   1,  8,  17,   8,  1;
%e A056152   1, 11,  42,  42,  11,  1;
%e A056152   1, 15,  91, 179,  91,  15,  1;
%e A056152   1, 19, 180, 633, 633, 180, 19, 1;
%e A056152   ...
%e A056152 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:
%e A056152 [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]
%e A056152 [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]
%e A056152 [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]
%e A056152 and
%e A056152 [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
%e A056152 [1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]
%e A056152 [1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].
%Y A056152 Columns k=1..6 are A000012, A024206, A055609, A055082, A055083, A055084.
%Y A056152 Row sums give A055192.
%Y A056152 See A122083 for another version of this triangle.
%Y A056152 Cf. A049312, A048194, A028657, A049311.
%K A056152 nonn,tabl
%O A056152 2,5
%A A056152 _Vladeta Jovovic_, Jul 29 2000