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A364281 Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length as an element.

%I A364281 #20 May 23 2025 04:12:30
%S A364281 1,1,1,4,10,48,252,1584,10800,93600,823680,8588160,93381120,
%T A364281 1158312960,14805504000,215028172800,3159494553600,51973589606400,
%U A364281 873152856576000,16058241239040000,300754643245056000,6159522883497984000,127439374149255168000
%N A364281 Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length as an element.
%H A364281 Alois P. Heinz, <a href="/A364281/b364281.txt">Table of n, a(n) for n = 0..450</a>
%H A364281 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A364281 Conjecture: a(n) ~ exp(1) * (n-1)!. - _Vaclav Kotesovec_, May 23 2025
%e A364281 a(3) = 4: (123), (132), (13)(2), (1)(23).
%e A364281 a(4) = 10: (1234), (1243), (1324), (1342), (1423), (1432), (124)(3),
%e A364281    (142)(3), (1)(234), (1)(243).
%p A364281 a:= proc(m) option remember; local b; b:=
%p A364281       proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A364281        `if`(n=0, p!*(m-p)!, b(n, i-1, p)+b(n-i, min(n-i, i-1), p-1)))
%p A364281       end: b(m$3)
%p A364281     end:
%p A364281 seq(a(n), n=0..24);
%t A364281 a[m_] := a[m] = Module[{b}, b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!*(m - p)!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p - 1]]]; b[m, m, m]];
%t A364281 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Oct 21 2023, after _Alois P. Heinz_ *)
%Y A364281 Cf. A327712, A364279, A364406.
%K A364281 nonn
%O A364281 0,4
%A A364281 _Alois P. Heinz_, Jul 17 2023