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Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.
5

%I #20 Jul 04 2024 18:34:04

%S 1,1,1,1,4,16,46,106,316,3564,27756,141516,556656,6678816,73015944,

%T 521124696,6144018336,75200767776,677927254176,4642387894944,

%U 75217104395136,1167068528384256,12348761954020416,97377968145352896,882819252604721664,66882151986021043200

%N Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.

%H Alois P. Heinz, <a href="/A374262/b374262.txt">Table of n, a(n) for n = 0..470</a>

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

%e a(4) = 4: (1)(2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

%p add(combinat[multinomial](n, i$i*j, n-i^2*j)*

%p b(n-i^2*j, i-1)*(i-1)!^(i*j)/(i*j)!, j=0..n/i^2))

%p end:

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

%p seq(a(n), n=0..25);

%Y Cf. A000704, A130268, A372545, A372579, A374292, A374320.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Jul 01 2024