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

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, 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, 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

Offset: 1

Author

W. Edwin Clark, Apr 18 2014

Keywords

• nonn
• hard

Comments

a(n) = 1 if every group of order n is abelian, that is, if n is in A051532.

Upper bounds are given by A000001 (number of groups of order n) and A018818 (number of partitions of n into divisors of n).

A077191 is an upper bound. - Eric M. Schmidt, Oct 16 2014

Examples

			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.
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1023
• Wikipedia, Conjugacy Class

Programs

GAP

a:=[];;
for n in [1..100] do
  P:=[];
  for i in [1..NumberSmallGroups(n)] do
   g:=SmallGroup(n,i);
   cc:=ConjugacyClasses(g);
   L:=List(cc,Size);
   Sort(L);
   Add(P,L);
   P:=Set(P);
  od;
  Add(a,Length(P));
od;
a;

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