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A372170 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles, 0 <= k <= binomial(n,3).

%I A372170 #17 Dec 29 2024 19:19:36
%S A372170 1,1,2,7,1,41,16,6,0,1,388,290,195,70,40,30,0,10,0,0,1,5789,6980,6910,
%T A372170 4560,3030,2292,1230,780,600,180,236,60,45,60,0,0,15,0,0,0,1,133501,
%U A372170 235270,313705,302505,260890,222509,174615,126780,102970,67165,50134,37485,20370,17990,11445,6552,4515,3570,1680,1785,154,735,455,140,0,105,105,0,0,0,21,0,0,0,0,1
%N A372170 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles, 0 <= k <= binomial(n,3).
%H A372170 Andrew Howroyd, <a href="/A372170/b372170.txt">Table of n, a(n) for n = 0..340</a> (rows 0..10)
%F A372170 Binomial transform of columns of A372167.
%e A372170 Triangle begins:
%e A372170      1
%e A372170      1
%e A372170      2
%e A372170      7    1
%e A372170     41   16    6    0    1
%e A372170    388  290  195   70   40   30    0   10    0    0    1
%e A372170    ...
%e A372170 For example, the T(4,1) = 16 graphs are:
%e A372170   12-13-23
%e A372170   12-14-24
%e A372170   13-14-34
%e A372170   23-24-34
%e A372170   12-13-14-23
%e A372170   12-13-14-24
%e A372170   12-13-14-34
%e A372170   12-13-23-24
%e A372170   12-13-23-34
%e A372170   12-14-23-24
%e A372170   12-14-24-34
%e A372170   12-23-24-34
%e A372170   13-14-23-34
%e A372170   13-14-24-34
%e A372170   13-23-24-34
%e A372170   14-23-24-34
%t A372170 cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}]&&MemberQ[y,{#[[1]],#[[3]]}]&&MemberQ[y,{#[[2]],#[[3]]}]&];
%t A372170 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[cys[#]]==k&]],{n,0,5},{k,0,Binomial[n,3]}]
%Y A372170 Row sums are A006125, covering A006129.
%Y A372170 Row lengths are A050407.
%Y A372170 Counting edges instead of triangles gives A084546, covering A054548.
%Y A372170 Column k = 0 is A213434, covering A372168.
%Y A372170 The unlabeled version is A263340.
%Y A372170 The covering case is A372167, unlabeled A372173.
%Y A372170 Column k = 1 is A372172, covering A372171.
%Y A372170 For all cycles (not just triangles) we have A372176, covering A372175.
%Y A372170 A001858 counts acyclic graphs, unlabeled A005195.
%Y A372170 A367867 counts non-choosable graphs, covering A367868.
%Y A372170 A372193 counts unicyclic graphs, unlabeled A236570, covering A372191.
%Y A372170 Cf. A000088, A000272, A002494, A006785, A053530, A062734, A121251, A372194.
%K A372170 nonn,tabf
%O A372170 0,3
%A A372170 _Gus Wiseman_, Apr 23 2024
%E A372170 a(42) onwards from _Andrew Howroyd_, Dec 29 2024