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)
1, 1, 1, 2, 10, 59, 363, 2491, 19661, 176536, 1767540, 19460671, 233578585, 3036411429, 42507793209, 637606959466, 10201702712738, 173429224591607, 3121728583605435, 59312852905363623, 1186257030934984061, 24911396924131631880, 548050726738352726108
Offset: 0
Keywords
Examples
a(3)=2 because we have (1)(2)(3) and (132).
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).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
Programs
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Maple
g := exp(1+z-exp(z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22); # second Maple program: a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)* binomial(n-1, j-1)*((j-1)!-`if`(j=1, 0, 1)), j=1..n)) end: seq(a(n), n=0..25); # Alois P. Heinz, Apr 13 2017 -
Mathematica
CoefficientList[Series[E^(1+x-E^x)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)
Formula
E.g.f.: exp(1+z-exp(z))/(1-z).
a(n) ~ n! * exp(2-exp(1)). - Vaclav Kotesovec, Oct 05 2013
a(n) = Sum_{k=0..1} A186754(n,k). - Alois P. Heinz, Dec 02 2021
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