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A181966
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Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.
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2
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0, 2, 12, 72, 480, 4320, 35280, 322560, 3265920, 39916800, 479001600, 6706022400, 93405312000, 1482030950400, 24845812992000, 418455797760000, 7469435990016000, 147254595231744000, 2919482409811968000, 63255452212592640000, 1430546380807864320000
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OFFSET
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1,2
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LINKS
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FORMULA
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PROG
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(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;
(PARI) a(n)={n!*sum(p=2, n, if(isprime(p), n\p))} \\ Andrew Howroyd, Jul 30 2018
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CROSSREFS
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Cf. A181954 for the number of such subgroups.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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