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A383878 Number of permutations of [n] with distinct cycle lengths whose GCD is 1.

%I A383878 #10 May 13 2025 11:16:31
%S A383878 0,1,0,3,8,50,264,2394,15840,158976,1490400,20124720,181543680,
%T A383878 3213905760,36459964800,602127540000,9045463311360,187660890063360,
%U A383878 2596164765465600,64849189355274240,1037566851245568000,24684232291242854400,498833466644833689600
%N A383878 Number of permutations of [n] with distinct cycle lengths whose GCD is 1.
%H A383878 Alois P. Heinz, <a href="/A383878/b383878.txt">Table of n, a(n) for n = 0..450</a>
%H A383878 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A383878 a(3) = 3: (1)(23), (13)(2), (12)(3).
%e A383878 a(4) = 8: (1)(234), (1)(243), (134)(2), (143)(2), (124)(3), (142)(3), (123)(4), (132)(4).
%p A383878 b:= proc(n, i, m) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, `if`(m=1, 1, 0),
%p A383878       b(n, i-1, m)+b(n-i, min(i-1, n-i), igcd(i, m))*binomial(n, i)*(i-1)!))
%p A383878     end:
%p A383878 a:= n-> b(n$2, 0):
%p A383878 seq(a(n), n=0..23);
%Y A383878 Cf. A079128, A382781.
%K A383878 nonn
%O A383878 0,4
%A A383878 _Alois P. Heinz_, May 13 2025