%I A084423
%S A084423 1,1,2,3,7,12,43,127,544,2361,11703,61690,351773,2126497,13639372,
%T A084423 92197523,655035769,4874404108,37893370473,306986431847,2586209749712,
%U A084423 22612848403571,204850732480285,1919652428481930,18581619724363401
%N A084423 Set partitions up to rotations.
%C A084423 Partitions of n objects distinct under the cyclic group, C_n. By comparison 
               the partition numbers (A000041) are the partitions distinct under 
               the symmetric group, S_n and the set partitions are those distinct 
               under the discrete group containing only the identity. - Franklin 
               T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 09 2008
%H A084423 Franklin T. Adams-Watters, <a href="b084423.txt">Table of n, a(n) for 
               n = 0..60</a>
%H A084423 Robert M. Dickau, <a href="http://mathforum.org/advanced/robertd/bell.html">
               Bell number diagrams</a>
%H A084423 Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/SetPartitionsUpToRotation/
               ">Set Partitions Up To Rotation</a>
%F A084423 a(p) = (Bell(p)+2*(p-1))/p for prime p; cf. A079609. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jul 04 2003
%F A084423 U(k,j) = 1 if k=0, else sum_{i=1}^k C(k-1,i-1) sum_{d|j} U(k-i,j)*d^{i-1}. 
               Then a(n) = (sum_{j-n} phi(j)*U(n/j,j))/n. (U(k,j) is the number 
               of partitions invariant under a permutation with k cycles of j objects 
               each.) - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 
               09 2008
%e A084423 Of the Bell(4)=15 set partitions of 4, only 7 remain distinct under rotation:
%e A084423 {{1,2,3,4}},
%e A084423 {{1}, {2,3,4}},
%e A084423 {{1,2}, {3,4}},
%e A084423 {{1,3}, {2,4}},
%e A084423 {{1}, {2}, {3,4}},
%e A084423 {{1}, {3}, {2,4}},
%e A084423 {{1}, {2}, {3}, {4}}}
%t A084423 <<DiscreteMath`Combinatorica`; shrink[n_Integer] := Union[ First[ Sort[ 
               NestList[Sort[Sort /@ ( #/.i_Integer:>Mod[i+1, n, 1])]&, #, n]]]& 
               /@ SetPartitions[n]]; Table[ Length[ shrink[k]], {k, 11}]
%o A084423 (PARI) U(k, j) = if(k==0,1,sum(i=1,k,binomial(k-1,i-1)*sumdiv(j,d,U(k-i,
               j)*d^(i-1)))) /* U is unoptimized; should remember previous values. 
               */ a(n) = sumdiv(n,j,eulerphi(j)*U(n\j,j))/n - Franklin T. Adams-Watters 
               (FrankTAW(AT)Netscape.net), Jun 09 2008
%Y A084423 Cf. A080107, A000110.
%Y A084423 Cf. A000041.
%Y A084423 Sequence in context: A143879 A056293 A056294 this_sequence A068134 A081256 
               A084955
%Y A084423 Adjacent sequences: A084420 A084421 A084422 this_sequence A084424 A084425 
               A084426
%K A084423 nonn,nice
%O A084423 0,3
%A A084423 Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jun 26 2003
%E A084423 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 27 2003
%E A084423 More terms from Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 09 2008