A124795 - OEIS

%I #35 Mar 05 2018 10:51:22

%S 1,1,1,1,2,3,6,1,3,8,24,6,120,30,20,1,720,15,5040,20,90,144,40320,10,

%T 40,840,15,90,362880,120,3628800,1,504,5760,420,45,39916800,45360,

%U 3360,40,479001600,630,6227020800,504,210,403200,87178291200,15,1260,280,25920

%N Number of permutations with given cycle structure, in the prime factorization order.

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

%H Gheorghe Coserea, <a href="/A124795/b124795.txt">Table of n, a(n) for n = 1..3000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Permutation cycle</a>

%F 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*...)

%t 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 *)

%o (PARI)

%o a(n) = {

%o   my(f=factor(n), fsz=matsize(f)[1],

%o      g=sum(k=1, fsz, primepi(f[k, 1]) * f[k, 2])!,

%o      h=prod(k=1, fsz, primepi(f[k, 1])^f[k, 2]));

%o   g/(prod(k=1, fsz, f[k, 2]!) * h);

%o };

%o vector(51, n, a(n)) \\ _Gheorghe Coserea_, Feb 02 2018; edited by _Max Alekseyev_, Feb 05 2018

%Y Cf. A000040.

%K nonn

%O 1,5

%A _Max Alekseyev_, Nov 07 2006