A317794 - OEIS

%I #18 Dec 04 2024 10:33:56

%S 1,1,2,8,180,612032,200253854316544,263735716028826427534807159537664,

%T 5609038300883759793482640992086670066760184863720423808367168537493504

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

%H 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 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.

%e Non-isomorphic representatives of the a(3) = 8 set-systems:

%e   0,

%e   {12}, {123},

%e   {12}{13}, {12}{123},

%e   {12}{13}{23}, {12}{13}{123},

%e   {12}{13}{23}{123}.

%t 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 Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]],Length[#]>1&]],Or[Length[#]==0,Union@@#==Range[Max@@Union@@#]]&]]],{n,4}]

%t (* second program *)

%t 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 The spanning case is A317795.

%Y Cf. A000088, A000612, A003180, A007716, A055621, A283877, A300913, A306005, A317533, A317757, A319876.

%K nonn

%O 0,3

%A _Gus Wiseman_, Aug 07 2018

%E More terms from _Gus Wiseman_, Dec 12 2018