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A003180 Number of equivalence classes of Boolean functions of n variables under action of symmetric group.
(Formerly M1265 N1405)
2, 4, 12, 80, 3984, 37333248, 25626412338274304, 67516342973185974328175690087661568, 2871827610052485009904013737758920847669809829897636746529411152822140928 (list; graph; refs; listen; history; text; internal format)



A003180(n-1) is the number of equivalence classes of Boolean functions of n variables from Post class F(8,inf) under action of symmetric group.

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

Also the number of unlabeled hypergraphs on n nodes [Qian]. - N. J. A. Sloane, May 12 2014

In the 1995 Encyclopedia of Integer Sequences this sequence appears twice, as both M1265 and M3458 (one entry began at n=0, the other at n=1).


M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 147.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

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

Qian, Jianguo. Enumeration of unlabeled uniform hypergraphs. Discrete Math. 326 (2014), 66--74. MR3188989. See Table 1, p. 71. - N. J. A. Sloane, May 12 2014

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).


Vladeta Jovovic, Table of n, a(n) for n = 0..11

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

Index entries for sequences related to Boolean functions


a(n) = Sum_{1*s_1+2*s_2+...=n} (fix A[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...)) where fix A[s_1, s_2, ...] = 2^Sum_{i>=1} ( Sum_{d|i} ( mu(i/d)*( 2^Sum_{j>=1} ( gcd(j, d)*s_j))))/i.



for n from 1 to 10 do

p:=partition(n): s:=0: for k from 1 to nops(p) do q:=convert(p[k], multiset): for i from 0 to n do a(i):=0: od:

  for i from 1 to nops(q) do a(q[i][1]):=q[i][2]: od:

  c:=1: ord:=1: for i from 1 to n do c:=c*a(i)!*i^a(i):ord:=lcm(ord, i): od: ss:=0:

  for i from 1 to ord do if ord mod i=0 then ss:=ss+phi(ord/i)*2^add(gcd(j, i)*a(j), j=1..n): fi: od:



printf(`%d `, n):

printf("%d ", s):

od: - Vladeta Jovovic, Sep 19 2006


a(n) = 2*A000612(n). Cf. A001146. Row sums of A052265.

Sequence in context: A141522 A114903 A038054 * A002080 A001206 A144295

Adjacent sequences:  A003177 A003178 A003179 * A003181 A003182 A003183




N. J. A. Sloane.


More terms from Vladeta Jovovic

Edited with formula by Christian G. Bower, Jan 08 2004



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Last modified October 6 22:33 EDT 2015. Contains 262347 sequences.