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A003181 Number of P-equivalence classes of nondegenerate Boolean functions of n variables.
(Formerly M0378)
12
2, 2, 8, 68, 3904, 37329264, 25626412300941056, 67516342973185974302549277749387264, 2871827610052485009904013737758920847602293486924450772201235462734479360 (list; graph; refs; listen; history; text; internal format)
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
S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971. [Annotated scans of a few pages]
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
Sequence in context: A181264 A354118 A224766 * A009616 A005615 A048617
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Christian Sievers, Jul 22 2016
Definition clarified by Ivo Timoteo, Mar 14 2017
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)