|
%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
|
