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A003180 Number of equivalence classes of Boolean functions of n variables under action of symmetric group.
(Formerly M1265 N1405)
30
2, 4, 12, 80, 3984, 37333248, 25626412338274304, 67516342973185974328175690087661568, 2871827610052485009904013737758920847669809829897636746529411152822140928 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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).
REFERENCES
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).
LINKS
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, Mathematics Stack Exchange, 2018.
FORMULA
a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[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).
EXAMPLE
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}}
{{},{1}}
{{1},{2}}
{{},{1,2}}
{{2},{1,2}}
{{},{1},{2}}
{{},{2},{1,2}}
{{1},{2},{1,2}}
{{},{1},{2},{1,2}}
(End)
MAPLE
with(numtheory):with(combinat):
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:
s:=s+2^(ss/ord)/c:
od:
printf(`%d `, n):
printf("%d ", s):
od: # Vladeta Jovovic, Sep 19 2006
MATHEMATICA
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-François 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+ *)
CROSSREFS
Twice A000612. Cf. A001146. Row sums of A052265.
Sequence in context: A141522 A114903 A038054 * A002080 A001206 A144295
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Sep 19 2006
Edited with formula by Christian G. Bower, Jan 08 2004
STATUS
approved

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Last modified April 21 13:26 EDT 2024. Contains 371870 sequences. (Running on oeis4.)