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A372175
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Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly 2k directed cycles of length > 2.
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12
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1, 0, 1, 3, 1, 19, 15, 0, 6, 0, 0, 0, 1, 155, 232, 15, 190, 0, 0, 70, 50, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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0,4
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COMMENTS
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A directed cycle in a simple (undirected) graph is a sequence of distinct vertices, up to rotation, such that there are edges between all consecutive elements, including the last and the first.
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LINKS
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EXAMPLE
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Triangle begins (zeros shown as dots):
1
.
1
3 1
19 15 . 6 ... 1
155 232 15 190 .. 70 50 .... 30 15 .......... 10 .............. 1
Row n = 4 counts the following graphs:
12,34 12,13,14,23 . 12,13,14,23,24 . . . 12,13,14,23,24,34
13,24 12,13,14,24 12,13,14,23,34
14,23 12,13,14,34 12,13,14,24,34
12,13,14 12,13,23,24 12,13,23,24,34
12,13,24 12,13,23,34 12,14,23,24,34
12,13,34 12,13,24,34 13,14,23,24,34
12,14,23 12,14,23,24
12,14,34 12,14,23,34
12,23,24 12,14,24,34
12,23,34 12,23,24,34
12,24,34 13,14,23,24
13,14,23 13,14,23,34
13,14,24 13,14,24,34
13,23,24 13,23,24,34
13,23,34 14,23,24,34
13,24,34
14,23,24
14,23,34
14,24,34
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MATHEMATICA
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cycles[g_]:=Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@g, {k}], Min@@#==First[#]&&And@@Table[MemberQ[Sort/@g, Sort[{#[[i]], #[[If[i==k, 1, i+1]]]}]], {i, k}]&], {k, 3, Length[g]}];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[cycles[#]]==2k&]], {n, 0, 5}, {k, 0, Length[cycles[Subsets[Range[n], {2}]]]/2}]
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CROSSREFS
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The non-covering version is A372176.
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KEYWORD
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nonn,more,tabf,new
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AUTHOR
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STATUS
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approved
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