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A124795
Number of permutations with given cycle structure, in the prime factorization order.
68
1, 1, 1, 1, 2, 3, 6, 1, 3, 8, 24, 6, 120, 30, 20, 1, 720, 15, 5040, 20, 90, 144, 40320, 10, 40, 840, 15, 90, 362880, 120, 3628800, 1, 504, 5760, 420, 45, 39916800, 45360, 3360, 40, 479001600, 630, 6227020800, 504, 210, 403200, 87178291200, 15, 1260, 280, 25920
OFFSET
1,5
COMMENTS
Number of permutations with k1 1-cycles, k2 2-cycles, ...
LINKS
Eric Weisstein's World of Mathematics, Permutation cycle
FORMULA
For n=p1^k1*p2^k2*... where 2=p1<p2<... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1^k1*2^k2*...)
MATHEMATICA
a[1] = 1; a[n_] := (f1 = FactorInteger[n]; rr = Range[PrimePi[f1[[-1, 1]]]]; f2 = {Prime[#], 0}& /@ rr; ff = Union[f1, f2] //. {b___, {p_, 0}, {p_, k_}, c___} -> {b, {p, k}, c}; kk = ff[[All, 2]]; (kk.rr)!/Times @@ (kk!)/Times @@ (rr^kk)); Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)
PROG
(PARI)
a(n) = {
my(f=factor(n), fsz=matsize(f)[1],
g=sum(k=1, fsz, primepi(f[k, 1]) * f[k, 2])!,
h=prod(k=1, fsz, primepi(f[k, 1])^f[k, 2]));
g/(prod(k=1, fsz, f[k, 2]!) * h);
};
vector(51, n, a(n)) \\ Gheorghe Coserea, Feb 02 2018; edited by Max Alekseyev, Feb 05 2018
CROSSREFS
Cf. A000040.
Sequence in context: A275732 A200594 A319191 * A368547 A346560 A084459
KEYWORD
nonn
AUTHOR
Max Alekseyev, Nov 07 2006
STATUS
approved