A364281 - OEIS

%I #18 Oct 21 2023 05:36:31

%S 1,1,1,4,10,48,252,1584,10800,93600,823680,8588160,93381120,

%T 1158312960,14805504000,215028172800,3159494553600,51973589606400,

%U 873152856576000,16058241239040000,300754643245056000,6159522883497984000,127439374149255168000

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

%H Alois P. Heinz, <a href="/A364281/b364281.txt">Table of n, a(n) for n = 0..450</a>

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

%e a(3) = 4: (123), (132), (13)(2), (1)(23).

%e a(4) = 10: (1234), (1243), (1324), (1342), (1423), (1432), (124)(3),

%e    (142)(3), (1)(234), (1)(243).

%p a:= proc(m) option remember; local b; b:=

%p       proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,

%p        `if`(n=0, p!*(m-p)!, b(n, i-1, p)+b(n-i, min(n-i, i-1), p-1)))

%p       end: b(m$3)

%p     end:

%p seq(a(n), n=0..24);

%t 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 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Oct 21 2023, after _Alois P. Heinz_ *)

%Y Cf. A327712, A364279, A364406.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jul 17 2023