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A332964
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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.
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0
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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, 46, 14, 6, 2, 1, 1, 10957, 1120, 194, 50, 15, 6, 2, 1, 1, 260494, 12675, 1219, 204, 51, 15, 6, 2, 1, 1, 11713772, 276758, 13099, 1254, 208, 52, 15, 6, 2, 1, 1
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OFFSET
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0,11
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COMMENTS
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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.
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REFERENCES
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C. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 166.
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) begins:
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, 46, 14, 6, 2, 1, 1;
...
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MATHEMATICA
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Needs["Combinatorica`"];
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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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