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A364406 Number of permutations of [n] such that the minimal element of each cycle is also its length.

%I A364406 #23 Dec 05 2023 19:45:17
%S A364406 1,1,0,1,0,0,6,6,0,0,720,2160,9360,19440,30240,3659040,21772800,
%T A364406 228614400,1632960000,11125900800,73025971200,1708337433600,
%U A364406 15442053580800,254260755302400,3318429200486400,46929444097536000,546974781889536000,7312714579602432000
%N A364406 Number of permutations of [n] such that the minimal element of each cycle is also its length.
%H A364406 Alois P. Heinz, <a href="/A364406/b364406.txt">Table of n, a(n) for n = 0..474</a>
%H A364406 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A364406 a(0) = 1: () the empty permutation.
%e A364406 a(1) = 1: (1).
%e A364406 a(3) = 1: (1)(23).
%e A364406 a(6) = 6: (1)(24)(356), (1)(24)(365), (1)(25)(346), (1)(25)(364),
%e A364406   (1)(26)(345), (1)(26)(354).
%e A364406 a(7) = 6: (1)(23)(4567), (1)(23)(4576), (1)(23)(4657), (1)(23)(4675),
%e A364406   (1)(23)(4756), (1)(23)(4765).
%p A364406 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
%p A364406       b(n, i-1)+`if`(2*i>n+1, 0, b(n-i, i-1)*binomial(n-i, i-1)*(i-1)!)))
%p A364406     end:
%p A364406 a:= n-> b(n$2):
%p A364406 seq(a(n), n=0..33);
%t A364406 b[n_, i_] := b[n, i] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1] + If[2*i > n + 1, 0, b[n - i, i - 1]*Binomial[n - i, i - 1]*(i - 1)!]]];
%t A364406 a[n_] := b[n, n];
%t A364406 Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Dec 05 2023, after _Alois P. Heinz_ *)
%Y A364406 Cf. A000142, A007837, A188431, A362362, A364207, A364277 A364279 A364281 A364283.
%K A364406 nonn
%O A364406 0,7
%A A364406 _Alois P. Heinz_, Jul 22 2023