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A372170
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Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles.
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15
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1, 1, 2, 7, 1, 41, 16, 6, 0, 1, 388, 290, 195, 70, 40, 30, 0, 10, 0, 0, 1, 5789, 6980, 6910, 4560, 3030, 2292, 1230, 780, 600, 180, 236, 60, 45, 60, 0, 0, 15, 0, 0, 0, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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Binomial transform of columns of A372167.
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EXAMPLE
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Triangle begins:
1
1
2
7 1
41 16 6 0 1
388 290 195 70 40 30 0 10 0 0 1
For example, the T(4,1) = 16 graphs are:
12-13-23
12-14-24
13-14-34
23-24-34
12-13-14-23
12-13-14-24
12-13-14-34
12-13-23-24
12-13-23-34
12-14-23-24
12-14-24-34
12-23-24-34
13-14-23-34
13-14-24-34
13-23-24-34
14-23-24-34
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MATHEMATICA
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cys[y_]:=Select[Subsets[Union@@y, {3}], MemberQ[y, {#[[1]], #[[2]]}]&&MemberQ[y, {#[[1]], #[[3]]}]&&MemberQ[y, {#[[2]], #[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[cys[#]]==k&]], {n, 0, 5}, {k, 0, Binomial[n, 3]}]
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CROSSREFS
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For all cycles (not just triangles) we have A372176, covering A372175.
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KEYWORD
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nonn,tabf,more,changed
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AUTHOR
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STATUS
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approved
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