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)
1, 0, 0, 1, 5, 23, 129, 894, 7202, 65085, 651263, 7161713, 85922825, 1116946192, 15637356864, 234562319757, 3753007054781, 63801128569995, 1148420035784849, 21819978138955622, 436399552962252082, 9164390639379582121, 201616594791853840063
Offset: 0
Keywords
Examples
a(4)=5 because we have (1432), (1342), (1423), (1243), and (1324).
a(5)=23 counts all cyclic permutations of {1,2,3,4,5}, except (12345).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
Programs
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Maple
g := exp(1-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)!-1), j=1..n)) end: seq(a(n), n=0..25); # Alois P. Heinz, Apr 13 2017 -
Mathematica
With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 23 2016 *)
Formula
E.g.f.: exp(1-exp(z))/(1-z).
a(n) ~ n! * exp(1-exp(1)) = 0.179374... * n!. - Vaclav Kotesovec, Mar 17 2014
Comments