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A332964 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n nodes with exactly k bipartite connected components, n >= 0, 0 <= k <= n.

%I A332964 #18 Mar 26 2020 23:18:43
%S A332964 1,0,1,0,1,1,1,1,1,1,3,4,2,1,1,16,9,5,2,1,1,96,37,13,6,2,1,1,812,162,
%T A332964 46,14,6,2,1,1,10957,1120,194,50,15,6,2,1,1,260494,12675,1219,204,51,
%U A332964 15,6,2,1,1,11713772,276758,13099,1254,208,52,15,6,2,1,1
%N A332964 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n nodes with exactly k bipartite connected components, n >= 0, 0 <= k <= n.
%C A332964 T(n,k) is the number of graphs on n nodes with incidence matrix of rank n-k, where the incidence matrix is defined as in Godsil-Royle reference below.
%D A332964 C. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 166.
%F A332964 G.f.: Product_{i>=1} (1/(1-x^i))^A157051(i)*(1/(1-y*x^i))^A005142(i).
%e A332964 Triangle T(n,k) begins:
%e A332964     1;
%e A332964     0,   1;
%e A332964     0,   1,   1;
%e A332964     1,   1,   1,  1;
%e A332964     3,   4,   2,  1,  1;
%e A332964    16,   9,   5,  2,  1, 1;
%e A332964    96,  37,  13,  6,  2, 1, 1;
%e A332964   812, 162,  46, 14,  6, 2, 1, 1;
%e A332964   ...
%t A332964 Needs["Combinatorica`"];
%t A332964 Table[Table[Count[Prepend[Flatten[Table[g = {n, k};b = GraphData[g,"IncidenceMatrix"]; {n - MatrixRank[b]}, {k,2, NumberOfGraphs[n]}]], n], i], {i, 0, n}], {n, 0,7}] // Grid
%Y A332964 Cf. A157051 (column k=0 for n>0), A000088 (row sums), A157015, A005142.
%K A332964 nonn,tabl
%O A332964 0,11
%A A332964 _Geoffrey Critzer_, Mar 04 2020