cp's OEIS Frontend

A372176 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.

%I A372176 #10 Aug 01 2024 10:33:22
%S A372176 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 A372176 0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A372176 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 A372176 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 A372176 Triangle begins (zeros shown as dots):
%e A372176    1
%e A372176    1
%e A372176    2
%e A372176    7 1
%e A372176    38 19 . 6 ... 1
%e A372176    291 317 15 220 .. 70 55 .... 30 15 ........ 10 ............... 1
%e A372176 The T(4,3) = 6 graphs:
%e A372176   12,13,14,23,24
%e A372176   12,13,14,23,34
%e A372176   12,13,14,24,34
%e A372176   12,13,23,24,34
%e A372176   12,14,23,24,34
%e A372176   13,14,23,24,34
%t A372176 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 A372176 Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cyc[#]]==2k&]], {n,0,4}, {k,0,Length[cyc[Subsets[Range[n],{2}]]]/2}]
%Y A372176 Column k = 0 is A001858 (unlabeled A005195), covering A105784.
%Y A372176 Row lengths are A002807 + 1.
%Y A372176 Row sums are A006125, unlabeled A000088.
%Y A372176 Counting edges instead of cycles gives A084546 (covering A054548), unlabeled A008406 (covering A370167).
%Y A372176 Counting triangles instead of cycles gives A372170 (covering A372167), unlabeled A263340 (covering A372173).
%Y A372176 The covering case is A372175.
%Y A372176 Column k = 1 is A372193 (covering A372195), unlabeled A236570.
%Y A372176 A006129 counts graphs, unlabeled A002494.
%Y A372176 A322661 counts covering loop-graphs, unlabeled A322700.
%Y A372176 Cf. A000272, A053530, A137916, A144958, A213434, A367863, A372168, A372169, A372171, A372172, A372191.
%K A372176 nonn,tabf,more
%O A372176 0,3
%A A372176 _Gus Wiseman_, Apr 25 2024