cp's OEIS Frontend

A241276 Number of partitions of n that come from sizes of conjugacy classes of groups of order n.

%I A241276 #21 Dec 13 2018 09:09:45
%S A241276 1,1,1,1,1,2,1,2,1,2,1,3,1,2,1,3,1,3,1,3,2,2,1,7,1,2,2,2,1,4,1,6,1,2,
%T A241276 1,6,1,2,2,5,1,6,1,2,1,2,1,13,1,3,1,3,1,7,2,5,2,2,1,9,1,2,2,16,1,4,1,
%U A241276 3,1,4,1,17,1,2,2,2,1,6,1,11,3,2,1,9,1,2,1,4,1,6,1,2,2,2,1,30,1,3,1,7
%N A241276 Number of partitions of n that come from sizes of conjugacy classes of groups of order n.
%C A241276 a(n) = 1 if every group of order n is abelian, that is, if n is in A051532.
%C A241276 Upper bounds are given by A000001 (number of groups of order n) and A018818 (number of partitions of n into divisors of n).
%C A241276 A077191 is an upper bound. - _Eric M. Schmidt_, Oct 16 2014
%H A241276 Eric M. Schmidt, <a href="/A241276/b241276.txt">Table of n, a(n) for n = 1..1023</a>
%H A241276 Wikipedia, <a href="https://en.wikipedia/wiki/Conjugacy_class">Conjugacy Class</a>
%e A241276 If n = 6 there are two groups of order 6: Z_6, all of whose conjugacy classes are of order 1 giving the partition [1,1,1,1,1,1] and S_6, which has three conjugacy classes whose sizes are 1, 2 and 3, giving the partition [1,2,3]. Hence a(6) = 2.
%o A241276 (GAP)
%o A241276 a:=[];;
%o A241276 for n in [1..100] do
%o A241276   P:=[];
%o A241276   for i in [1..NumberSmallGroups(n)] do
%o A241276    g:=SmallGroup(n,i);
%o A241276    cc:=ConjugacyClasses(g);
%o A241276    L:=List(cc,Size);
%o A241276    Sort(L);
%o A241276    Add(P,L);
%o A241276    P:=Set(P);
%o A241276   od;
%o A241276   Add(a,Length(P));
%o A241276 od;
%o A241276 a;
%o A241276 (GAP) a := function(n) local i, p, P; P := []; for i in [1..NrSmallGroups(n)] do p := List(ConjugacyClasses(SmallGroup(n,i)), Size); Sort(p); MakeImmutable(p); AddSet(P, p); od; return Length(P); end; # _Eric M. Schmidt_, Oct 16 2014
%K A241276 nonn,hard
%O A241276 1,6
%A A241276 _W. Edwin Clark_, Apr 18 2014