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

1,5

COMMENTS

Number of permutations with k1 1-cycles, k2 2-cycles, ...

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..3000

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-Fran├žois 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 * A084459 A093095 A260611

Adjacent sequences:  A124792 A124793 A124794 * A124796 A124797 A124798

KEYWORD

nonn

AUTHOR

Max Alekseyev, Nov 07 2006

STATUS

approved

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Last modified October 23 12:04 EDT 2019. Contains 328345 sequences. (Running on oeis4.)