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A364283 Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length different from its own as an element.

%I A364283 #18 Jul 23 2023 17:04:02
%S A364283 1,0,0,1,2,12,60,408,2640,24480,208080,2262960,23950080,307359360,
%T A364283 3835641600,57400358400,825160089600,13909727462400,229664981145600,
%U A364283 4310966499840000,79428141112320000,1658163790483200000,33795850208440320000,770528520983789568000
%N A364283 Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length different from its own as an element.
%H A364283 Alois P. Heinz, <a href="/A364283/b364283.txt">Table of n, a(n) for n = 0..450</a>
%H A364283 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A364283 a(3) = 1: (13)(2).
%e A364283 a(4) = 2: (124)(3), (142)(3).
%e A364283 a(5) = 12: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4),
%e A364283    (124)(35), (142)(35), (125)(34), (152)(34), (13)(245), (13)(254).
%p A364283 f:= proc(n) option remember; `if`(n<2, 1-n, (n-1)*(f(n-1)+f(n-2))) end:
%p A364283 a:= proc(m) option remember; local b; b:=
%p A364283       proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A364283        `if`(n=0, p!*f(m-p), b(n, i-1, p)+b(n-i, min(n-i, i-1), p-1)))
%p A364283       end: b(m$3)
%p A364283     end:
%p A364283 seq(a(n), n=0..24);
%Y A364283 Cf. A000009, A000166, A007838, A362362, A364277, A364279, A364281, A364282, A364406.
%K A364283 nonn
%O A364283 0,5
%A A364283 _Alois P. Heinz_, Jul 17 2023