A003181 Number of P-equivalence classes of nondegenerate Boolean functions of n variables. (Formerly M0378)

2, 2, 8, 68, 3904, 37329264, 25626412300941056, 67516342973185974302549277749387264, 2871827610052485009904013737758920847602293486924450772201235462734479360

Offset

0,1

Comments

Also the number of non-isomorphic sets of subsets of {1..n} with union {1..n}. - Gus Wiseman, Aug 05 2019

References

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.

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

Links

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

S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971. [Annotated scans of a few pages]

Index entries for sequences related to Boolean functions

Formula

a(n) = A003180(n) - A003180(n-1), for n >= 1. - Christian Sievers, Jul 22 2016

a(n) = 2 * A055621(n). - Gus Wiseman, Aug 05 2019

Example

From Gus Wiseman, Aug 05 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{1},{2}}
                  {{},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

Maple

b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
      h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
    end:
a:= n-> `if`(n=0, 2, b(n$2, [])-b(n-1$2, [])):
seq(a(n), n=0..8);  # Alois P. Heinz, Aug 14 2019

Mathematica

b[n_, i_, l_] := If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
a[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]];
a /@ Range[0, 8] (* Jean-François Alcover, Apr 11 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000371, A001146, A003180, A003465, A055621, A007537, A326881.

Sequence in context: A181264 A354118 A224766 * A009616 A005615 A048617

Adjacent sequences: A003178 A003179 A003180 * A003182 A003183 A003184

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian Sievers, Jul 22 2016

Definition clarified by Ivo Timoteo, Mar 14 2017

Status

approved