The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

The number of unlabeled hypergraphs with empty hyperedges allowed on n nodes. Compare with A000612 where empty hyperedges are not allowed. - Michael Somos, Feb 15 2019

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.

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

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.

Toru Ishihara, Enumeration of hypergraphs, European Journal of Combinatorics, Volume 22, Issue 4, May 2001.

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

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

Marko Riedel, Cycle indices for the enumeration of non-isomorphic hypergraphs, Math StackExchange, 2018.

Marko Riedel, Implementation of the Ishihara algorithm for cycle indices of the action of the symmetric group S_n on sets of subsets of an n-set.

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.

a(n) = 2 * A000612(n).


From Gus Wiseman, Aug 05 2019: (Start)

Non-isomorphic representatives of the a(0) = 2 through a(2) = 12 sets of subsets:

  {}    {}        {}

  {{}}  {{}}      {{}}

        {{1}}     {{1}}

        {{},{1}}  {{1,2}}












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_] := 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]}];

a /@ Range[0, 11] (* Jean-Fran├žois Alcover, Feb 19 2020, after Alois P. Heinz in A000612 *)

fix[s_] := 2^Sum[Sum[MoebiusMu[i/d] 2^Sum[GCD[j, d] s[j], {j, Keys[s]}], {d, Divisors[i]}]/i, {i, LCM @@ Keys[s]}];

a[0] = 2;

a[n_] := Sum[fix[s]/Product[j^s[j] s[j]!, {j, Keys[s]}], {s, Counts /@ IntegerPartitions[n]}];

Table[a[n], {n, 0, 8}]

(* Andrey Zabolotskiy, Mar 24 2020, after Christian G. Bower's formula; requires Mathematica 10+ *)


Twice A000612. Cf. A001146. Row sums of A052265.

Cf. A003181, A055621.

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, Sep 19 2006

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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 5 14:42 EDT 2021. Contains 346469 sequences. (Running on oeis4.)