

A241276


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


1



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
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OFFSET

1,6


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


LINKS

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


EXAMPLE

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.


PROG

(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


CROSSREFS

Sequence in context: A087942 A327925 A320012 * A325759 A292286 A099042
Adjacent sequences: A241273 A241274 A241275 * A241277 A241278 A241279


KEYWORD

nonn,hard


AUTHOR

W. Edwin Clark, Apr 18 2014


STATUS

approved



