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)<... .

1, 0, 0, 1, 5, 23, 129, 894, 7202, 65085, 651263, 7161713, 85922825, 1116946192, 15637356864, 234562319757, 3753007054781, 63801128569995, 1148420035784849, 21819978138955622, 436399552962252082, 9164390639379582121, 201616594791853840063

Offset

0,5

Comments

a(n) = A186754(n,0).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

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

Example

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).

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 *)

Crossrefs

Cf. A186754, A186756, A186757, A186758, A186759, A186760.

Sequence in context: A200028 A020032 A293088 * A009321 A078509 A239820

Adjacent sequences: A186752 A186753 A186754 * A186756 A186757 A186758

Keyword

nonn

Author

Emeric Deutsch, Feb 26 2011

Status

approved