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A186755 Number of permutations of {1,2,...,n} having no increasing cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) .

%I A186755 #15 Apr 13 2017 10:22:04
%S A186755 1,0,0,1,5,23,129,894,7202,65085,651263,7161713,85922825,1116946192,
%T A186755 15637356864,234562319757,3753007054781,63801128569995,
%U A186755 1148420035784849,21819978138955622,436399552962252082,9164390639379582121,201616594791853840063
%N A186755 Number of permutations of {1,2,...,n} having no increasing cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... .
%C A186755 a(n) = A186754(n,0).
%H A186755 Alois P. Heinz, <a href="/A186755/b186755.txt">Table of n, a(n) for n = 0..450</a>
%F A186755 E.g.f.: exp(1-exp(z))/(1-z).
%F A186755 a(n) ~ n! * exp(1-exp(1)) = 0.179374... * n!. - _Vaclav Kotesovec_, Mar 17 2014
%e A186755 a(4)=5 because we have (1432), (1342), (1423), (1243), and (1324).
%e A186755 a(5)=23 counts all cyclic permutations of {1,2,3,4,5}, except (12345).
%p A186755 g := exp(1-exp(z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
%p A186755 # second Maple program:
%p A186755 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A186755       a(n-j)*binomial(n-1, j-1)*((j-1)!-1), j=1..n))
%p A186755     end:
%p A186755 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 13 2017
%t A186755 With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 23 2016 *)
%Y A186755 Cf. A186754, A186756, A186757, A186758, A186759, A186760.
%K A186755 nonn
%O A186755 0,5
%A A186755 _Emeric Deutsch_, Feb 26 2011