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

0,3

Comments

a(0) = 1 by convention.

Links

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

Wikipedia, Permutation

Formula

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

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

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);

Crossrefs

Cf. A000035, A000142, A364967.

Sequence in context: A177474 A372987 A177480 * A089179 A177483 A004104

Adjacent sequences: A365226 A365227 A365228 * A365230 A365231 A365232

Keyword

nonn

Author

Alois P. Heinz, Aug 27 2023

Status

approved