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A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

%I A370167 #11 Feb 19 2024 14:18:12
%S A370167 1,0,0,1,0,0,1,1,0,0,1,2,2,1,1,0,0,0,1,4,5,5,4,2,1,1,0,0,0,1,3,9,15,
%T A370167 20,22,20,14,9,5,2,1,1,0,0,0,0,1,6,20,41,73,110,133,139,126,95,64,40,
%U A370167 21,10,5,2,1,1,0,0,0,0,1,3,15,50,124,271,515,832,1181,1460,1581,1516,1291,970,658,400,220,114,56,24,11,5,2,1,1
%N A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.
%H A370167 Andrew Howroyd, <a href="/A370167/b370167.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)
%e A370167 Triangle begins:
%e A370167   1
%e A370167   0
%e A370167   0  1
%e A370167   0  0  1  1
%e A370167   0  0  1  2  2  1  1
%e A370167   0  0  0  1  4  5  5  4  2  1  1
%e A370167   0  0  0  1  3  9 15 20 22 20 14  9  5  2  1  1
%t A370167 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
%t A370167 Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}],{k}],Union@@#==Range[n]&]]], {n,0,5},{k,0,Binomial[n,2]}]
%o A370167 (PARI) \\ G(n) defined in A008406.
%o A370167 row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n,2)+1)}
%o A370167 { for(n=0, 7, print(row(n))) } \\ _Andrew Howroyd_, Feb 19 2024
%Y A370167 Column sums are A000664.
%Y A370167 Row sums are A002494.
%Y A370167 This is the covering case of A008406, labeled A084546.
%Y A370167 The labeled version is A054548, row sums A006129, column sums A121251.
%Y A370167 The connected case is A054924, row sums A001349, column sums A002905.
%Y A370167 The labeled connected case is A062734, with loops A369195.
%Y A370167 The connected case with loops is A283755, row sums A054921.
%Y A370167 The labeled version w/ loops is A369199, row sums A322661, col sums A173219.
%Y A370167 Cf. A000666, A006125, A006649, A054547, A066383, A322700.
%K A370167 nonn,tabf
%O A370167 0,12
%A A370167 _Gus Wiseman_, Feb 15 2024
%E A370167 a(42) onwards from _Andrew Howroyd_, Feb 19 2024