A232248 Denominators of the expected length of a random cycle in a random permutation.

1, 2, 12, 24, 720, 1440, 60480, 4480, 3628800, 1036800, 479001600, 958003200, 2615348736000, 172204032, 2414168064000, 62768369664000, 2462451425280000, 9146248151040000, 51090942171709440000, 136216903680000, 33720021833328230400000, 67440043666656460800000

Offset: 1

Author

Geoffrey Critzer, Nov 21 2013

Keywords

• nonn
• frac

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..250

Crossrefs

Numerators are A232193.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j
      *b(n-i*j, i-1) *x^j, j=0..n/i))))
    end:
a:= n->denom((p->add(coeff(p, x, i)/i, i=1..n))(b(n$2))/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 21 2013
# second Maple program:
a:= n-> denom(add(abs(combinat[stirling1](n, i))/i, i=1..n)/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 23 2013

Mathematica

Table[Denominator[Total[Map[Total[#]!/Product[#[[i]],{i,1,Length[#]}]/Apply[Times,Table[Count[#,k]!,{k,1,Max[#]}]]/(Total[#]-1)!/Length[#]&,Partitions[n]]]],{n,1,25}]

Formula

a(n) = Denominator( 1/(n-1)! * Sum_{i=1..n} A132393(n,i)/i ). - Alois P. Heinz, Nov 23 2013

a(n) = denominator(Sum_{k=0..n} A002657(k)/A091137(k)) (conjectured). - Michel Marcus, Jul 19 2019