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A317794 Number of non-isomorphic set-systems on n vertices with no singletons.

%I A317794 #21 Apr 19 2025 18:06:09
%S A317794 1,1,2,8,180,612032,200253854316544,263735716028826427534807159537664,
%T A317794 5609038300883759793482640992086670066760184863720423808367168537493504
%N A317794 Number of non-isomorphic set-systems on n vertices with no singletons.
%H A317794 Loïc Foissy, <a href="https://arxiv.org/abs/2304.00810">Hopf algebraic structures on hypergraphs and multi-complexes</a>, arXiv:2304.00810 [math.CO], 2023.
%H A317794 Peter H. van der Kamp, <a href="https://arxiv.org/abs/2411.18264">Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials</a>, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.
%F A317794 a(n) = A000616(n) - A000370(n). - _Tilman Piesk_, Apr 14 2025
%e A317794 Non-isomorphic representatives of the a(3) = 8 set-systems:
%e A317794   0,
%e A317794   {12}, {123},
%e A317794   {12}{13}, {12}{123},
%e A317794   {12}{13}{23}, {12}{13}{123},
%e A317794   {12}{13}{23}{123}.
%t A317794 sysnorm[{}] := {};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
%t A317794 Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]],Length[#]>1&]],Or[Length[#]==0,Union@@#==Range[Max@@Union@@#]]&]]],{n,4}]
%t A317794 (* second program *)
%t A317794 Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Subsets[Range[n],{2,n}]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length]/n!,{prm,Permutations[Range[n]]}],{n,6}] (* _Gus Wiseman_, Dec 12 2018 *)
%Y A317794 The spanning case is A317795.
%Y A317794 Cf. A000088, A000612, A003180, A007716, A055621, A283877, A300913, A306005, A317533, A317757, A319876, A000616, A000370.
%K A317794 nonn
%O A317794 0,3
%A A317794 _Gus Wiseman_, Aug 07 2018
%E A317794 More terms from _Gus Wiseman_, Dec 12 2018