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Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices with exactly 2k directed cycles of length > 2.
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%I #10 Aug 01 2024 10:33:22

%S 1,1,2,7,1,38,19,0,6,0,0,0,1,291,317,15,220,0,0,70,55,0,0,0,0,30,15,0,

%T 0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1

%N Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices with exactly 2k directed cycles of length > 2.

%C 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.

%e Triangle begins (zeros shown as dots):

%e 1

%e 1

%e 2

%e 7 1

%e 38 19 . 6 ... 1

%e 291 317 15 220 .. 70 55 .... 30 15 ........ 10 ............... 1

%e The T(4,3) = 6 graphs:

%e 12,13,14,23,24

%e 12,13,14,23,34

%e 12,13,14,24,34

%e 12,13,23,24,34

%e 12,14,23,24,34

%e 13,14,23,24,34

%t cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}], And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&], {k,3,Length[y]}],Min@@#==First[#]&];

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cyc[#]]==2k&]], {n,0,4}, {k,0,Length[cyc[Subsets[Range[n],{2}]]]/2}]

%Y Column k = 0 is A001858 (unlabeled A005195), covering A105784.

%Y Row lengths are A002807 + 1.

%Y Row sums are A006125, unlabeled A000088.

%Y Counting edges instead of cycles gives A084546 (covering A054548), unlabeled A008406 (covering A370167).

%Y Counting triangles instead of cycles gives A372170 (covering A372167), unlabeled A263340 (covering A372173).

%Y The covering case is A372175.

%Y Column k = 1 is A372193 (covering A372195), unlabeled A236570.

%Y A006129 counts graphs, unlabeled A002494.

%Y A322661 counts covering loop-graphs, unlabeled A322700.

%Y Cf. A000272, A053530, A137916, A144958, A213434, A367863, A372168, A372169, A372171, A372172, A372191.

%K nonn,tabf,more

%O 0,3

%A _Gus Wiseman_, Apr 25 2024