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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.
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%I #12 Aug 28 2023 11:37:32

%S 1,1,2,6,20,85,382,2219,13624,100293,811914,7594015,74507490,

%T 862987151,10327793088,139175089681,1966790900028,30983071424315,

%U 496696984054286,8925920862110603,162253809011669330,3228438870635420315,65677024568975412036,1448358661756969370985

%N 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.

%C a(0) = 1 by convention.

%H Alois P. Heinz, <a href="/A365229/b365229.txt">Table of n, a(n) for n = 0..275</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

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

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

%p b:= proc(n, l, m) option remember; `if`(n=0, 1/(m-l)!, add((j-1)!

%p *b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1), j=1..n))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..23);

%Y Cf. A000035, A000142, A364967.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 27 2023