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A186758 Number of permutations of {1,2,...,n} with no increasing cycles of length >=2. 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 A186758 #14 Dec 02 2021 17:46:01
%S A186758 1,1,1,2,10,59,363,2491,19661,176536,1767540,19460671,233578585,
%T A186758 3036411429,42507793209,637606959466,10201702712738,173429224591607,
%U A186758 3121728583605435,59312852905363623,1186257030934984061,24911396924131631880,548050726738352726108
%N A186758 Number of permutations of {1,2,...,n} with no increasing cycles of length >=2. 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 A186758 a(n) = A186757(n,0).
%H A186758 Alois P. Heinz, <a href="/A186758/b186758.txt">Table of n, a(n) for n = 0..450</a>
%F A186758 E.g.f.: exp(1+z-exp(z))/(1-z).
%F A186758 a(n) ~ n! * exp(2-exp(1)). - _Vaclav Kotesovec_, Oct 05 2013
%F A186758 a(n) = Sum_{k=0..1} A186754(n,k). - _Alois P. Heinz_, Dec 02 2021
%e A186758 a(3)=2 because we have (1)(2)(3) and (132).
%e A186758 a(4)=10 because we have (1)(2)(34), (1)(243), (132)(4), (142)(3), (143)(2), and the 5 cyclic permutations of {1,2,3,4} different from (1234).
%p A186758 g := exp(1+z-exp(z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
%p A186758 # second Maple program:
%p A186758 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A186758       binomial(n-1, j-1)*((j-1)!-`if`(j=1, 0, 1)), j=1..n))
%p A186758     end:
%p A186758 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 13 2017
%t A186758 CoefficientList[Series[E^(1+x-E^x)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 05 2013 *)
%Y A186758 Cf. A186754, A186755, A186756, A186757, A186759, A186760.
%K A186758 nonn
%O A186758 0,4
%A A186758 _Emeric Deutsch_, Feb 26 2011