A181966 Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.

0, 2, 12, 72, 480, 4320, 35280, 322560, 3265920, 39916800, 479001600, 6706022400, 93405312000, 1482030950400, 24845812992000, 418455797760000, 7469435990016000, 147254595231744000, 2919482409811968000, 63255452212592640000, 1430546380807864320000

Offset: 1

Author

Olivier Gérard, Apr 04 2012

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Math.stackexchange.com, Normalizer of the cyclic group in S_n

Crossrefs

Cf. A181954 for the number of such subgroups.

Cf. A013939, A181967.

Programs

GAP

List([1..7], n->Sum(Filtered( ConjugacyClassesSubgroups( SymmetricGroup(n)), x->IsPrime( Size( Representative(x))) ), x->Size(x)*Size( Normalizer( SymmetricGroup(n), Representative(x))) )); # Andrew Howroyd, Jul 30 2018

GAP

a:=function(n) local total, perm, g, p, k;
  total:= 0; g:= SymmetricGroup(n);
  for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n,p)] do
     perm:=PermList(List([0..p*k-1], i->i - (i mod p) + ((i + 1) mod p) + 1));
     total:=total + Size(Normalizer(g, perm)) * Factorial(n) / (p^k * (p-1) * Factorial(k) * Factorial(n-k*p));
  od; od;
  return total;
end; # Andrew Howroyd, Jul 30 2018

PARI

a(n)={n!*sum(p=2, n, if(isprime(p), n\p))} \\ Andrew Howroyd, Jul 30 2018

Formula

a(n) = n! * A013939(n). - Andrew Howroyd, Jul 30 2018

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jul 30 2018