A365229 Sum over all k of 1/k! times the number of permutations of [n] for which the difference between the longest and the shortest cycle length is k.

1, 1, 2, 6, 20, 85, 382, 2219, 13624, 100293, 811914, 7594015, 74507490, 862987151, 10327793088, 139175089681, 1966790900028, 30983071424315, 496696984054286, 8925920862110603, 162253809011669330, 3228438870635420315, 65677024568975412036, 1448358661756969370985

Offset: 0

Author

Alois P. Heinz, Aug 27 2023

Keywords

• nonn

Comments

a(0) = 1 by convention.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..275
• Wikipedia, Permutation

Crossrefs

Cf. A000035, A000142, A364967.

Programs

Maple

b:= proc(n, l, m) option remember; `if`(n=0, 1/(m-l)!, add((j-1)!
      *b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..23);

Formula

a(n) = Sum_{k=0..max(0,n-2)} A364967(n,k)/k!.

a(n) mod 2 = A000035(n) for n>=4.