A241276 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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;

(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: A378218 A359508 A352825 * A325759 A292286 A341596

Adjacent sequences: A241273 A241274 A241275 * A241277 A241278 A241279

KEYWORD

nonn,hard

AUTHOR

W. Edwin Clark, Apr 18 2014

STATUS

approved