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%I A000612 M1712 N0677 #54 Oct 30 2023 13:25:11
%S A000612 1,2,6,40,1992,18666624,12813206169137152,
%T A000612 33758171486592987164087845043830784,
%U A000612 1435913805026242504952006868879460423834904914948818373264705576411070464
%N A000612 Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.
%C A000612 Also number of nonisomorphic sets of nonempty subsets of an n-set.
%C A000612 Equivalently, number of nonisomorphic fillings of a Venn diagram of n sets. - _Joerg Arndt_, Mar 24 2020
%C A000612 Number of hypergraphs on n unlabeled nodes. - _Charles R Greathouse IV_, Apr 06 2021
%D A000612 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 153.
%D A000612 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 Table 2.3.2. - Row 5.
%D A000612 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000612 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000612 Alois P. Heinz, <a href="/A000612/b000612.txt">Table of n, a(n) for n = 0..12</a>
%H A000612 M. A. Harrison, <a href="http://dx.doi.org/10.1109/PGEC.1963.263656">The number of equivalence classes of Boolean functions under groups containing negation</a>, IEEE Trans. Electron. Comput. 12 (1963), 559-561.
%H A000612 M. A. Harrison, <a href="/A000370/a000370.pdf">The number of equivalence classes of Boolean functions under groups containing negation</a>, IEEE Trans. Electron. Comput. 12 (1963), 559-561. [Annotated scanned copy]
%H A000612 Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, <a href="https://arxiv.org/abs/2310.15668">Hypergraph Motifs and Their Extensions Beyond Binary</a>, arXiv:2310.15668 [cs.SI], 2023.
%H A000612 Wikipedia, <a href="https://en.wikipedia.org/wiki/Venn_diagram">Venn diagram</a>
%H A000612 <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%F A000612 a(n) = A003180(n)/2.
%e A000612 Non-isomorphic representatives of the a(2) = 6 set-systems are 0, {1}, {12}, {1}{2}, {1}{12}, {1}{2}{12}. - _Gus Wiseman_, Aug 07 2018
%p A000612 a:= n-> add(1/(p-> mul((c-> j^c*c!)(coeff(p, x, j)), j=1..degree(p)))(
%p A000612         add(x^i, i=l))*2^((w-> add(mul(2^igcd(t, l[i]), i=1..nops(l)),
%p A000612         t=1..w)/w)(ilcm(l[]))), l=combinat[partition](n))/2:
%p A000612 seq(a(n), n=0..9);  # _Alois P. Heinz_, Aug 12 2019
%t A000612 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 A000612 Table[Length[Union[sysnorm/@Subsets[Rest[Subsets[Range[n]]]]]],{n,4}] (* _Gus Wiseman_, Aug 07 2018 *)
%t A000612 a[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l=={}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}]/2;
%t A000612 a /@ Range[0, 9] (* _Jean-François Alcover_, Feb 04 2020, after _Alois P. Heinz_ *)
%Y A000612 a(n) = A003180(n)/2.
%Y A000612 Cf. A007716, A055621, A058891, A283877, A300913, A306005, A317533, A317757.
%K A000612 nonn,easy,nice,core
%O A000612 0,2
%A A000612 _N. J. A. Sloane_
%E A000612 More terms from _Vladeta Jovovic_, Feb 23 2000

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