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A181967
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Sum of the sizes of the normalizers of all prime order cyclic subgroups of the alternating group A_n.
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3
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0, 0, 3, 24, 180, 1440, 12600, 120960, 1270080, 14515200, 179625600, 2634508800, 37362124800, 566658892800, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 25545471085854720000, 587545834974658560000, 13488008733331292160000
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OFFSET
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1,3
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COMMENTS
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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( AlternatingGroup(n)), x->IsPrime( Size( Representative(x))) ), x->Size(x)*Size( Normalizer( AlternatingGroup(n), Representative(x))) )); # Andrew Howroyd, Jul 30 2018
(GAP)
a:=function(n) local total, perm, g, p, k;
total:= 0; g:= AlternatingGroup(n);
for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n, p)] do
if p>2 or IsEvenInt(k) then
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));
fi;
od; od;
return total;
(PARI) a(n)={n!*sum(p=2, n, if(isprime(p), if(p==2, n\4, n\p)))/2} \\ Andrew Howroyd, Jul 30 2018
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CROSSREFS
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Cf. A181951 for the number of such subgroups.
Cf. A181966 is the symmetric group case.
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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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