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A000612 Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.
(Formerly M1712 N0677)
1, 2, 6, 40, 1992, 18666624, 12813206169137152, 33758171486592987164087845043830784, 1435913805026242504952006868879460423834904914948818373264705576411070464 (list; graph; refs; listen; history; text; internal format)



Also number of nonisomorphic sets of nonempty subsets of an n-set.

Equivalently, number of nonisomorphic fillings of a Venn diagram of n sets. - Joerg Arndt, Mar 24 2020


M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 153.

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 Table 2.3.2. - Row 5.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Alois P. Heinz, Table of n, a(n) for n = 0..12

M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561.

M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561. [Annotated scanned copy]

Wikipedia, Venn diagram

Index entries for sequences related to Boolean functions


a(n) = A003180(n)/2.


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


a:= n-> add(1/(p-> mul((c-> j^c*c!)(coeff(p, x, j)), j=1..degree(p)))(

        add(x^i, i=l))*2^((w-> add(mul(2^igcd(t, l[i]), i=1..nops(l)),

        t=1..w)/w)(ilcm(l[]))), l=combinat[partition](n))/2:

seq(a(n), n=0..9);  # Alois P. Heinz, Aug 12 2019


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]]}]])]];

Table[Length[Union[sysnorm/@Subsets[Rest[Subsets[Range[n]]]]]], {n, 4}] (* Gus Wiseman, Aug 07 2018 *)

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;

a /@ Range[0, 9] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz *)


a(n) = A003180(n)/2.

Cf. A007716, A055621, A058891, A283877, A300913, A306005, A317533, A317757.

Sequence in context: A135755 A051185 A118623 * A319633 A326268 A096138

Adjacent sequences:  A000609 A000610 A000611 * A000613 A000614 A000615




N. J. A. Sloane


More terms from Vladeta Jovovic, Feb 23 2000



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Last modified May 30 14:49 EDT 2020. Contains 334726 sequences. (Running on oeis4.)