OFFSET
1,3
COMMENTS
In other words, if the smallest cycle in the n-permutation has length k then no other cycle in the permutation has length k.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..449
FORMULA
E.g.f.: Sum_{k>=1} x^k/k * exp(-Sum_{i=1..k}x^i/i)/(1-x).
EXAMPLE
a(4) = 14 because we have 14 such permutations of {1,2,3,4} shown in cycle notation: {{1}, {3,4,2}}, {{1}, {4,3,2}}, {{2,3,1}, {4}}, {{2,3,4,1}}, {{2,4,3,1}}, {{2,4,1}, {3}}, {{3,2,1}, {4}}, {{3,4,2,1}}, {{3,4,1}, {2}}, {{3,2,4,1}}, {{4,3,2,1}}, {{4,2,1}, {3}}, {{4,3,1}, {2}}, {{4,2,3,1}}.
MAPLE
with(combinat):
b:= proc(n, i) option remember;
`if`(i<1, 0, `if`(n=i, (i-1)!, 0) +add(b(n-i*j, i-1)*
multinomial(n, n-i*j, i$j)/j!*(i-1)!^j, j=0..(n-1)/i))
end:
a:= n-> b(n$2):
seq(a(n), n=1..25); # Alois P. Heinz, Sep 07 2020
MATHEMATICA
nn=20; Drop[Range[0, nn]! CoefficientList[Series[Sum[x^k/k Exp[-Sum[x^i/i, {i, 1, k}]]/(1-x), {k, 1, nn}], {x, 0, nn}], x], 1]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Apr 01 2013
STATUS
approved