A129135 Number of permutations of [n] with exactly 5 fixed points.

1, 0, 21, 112, 1134, 11088, 122430, 1468368, 19090071, 267258992, 4008887883, 64142201760, 1090417436108, 19627513841376, 372922762997772, 7458455259939936, 156627560458759005, 3445806330092671776, 79253545592131484497, 1902085094211155585424

Offset: 5

Author

Zerinvary Lajos, May 25 2007

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 5..200
• FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
• Index entries for sequences related to permutations with fixed points

Crossrefs

Cf. A008290, A008291, A170942.

Programs

Maple

a:=n->sum((n-1)!*sum((-1)^k/(k-4)!, j=0..n-1), k=4..n-1)/5!: seq(a(n), n=5..24);
x:='x'; G(x):=exp(-x)/(1-x)*(x^5/5!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=5..24); # Zerinvary Lajos, Apr 03 2009
a:= n-> simplify(pochhammer(6, n-5)*GAMMA(n-4, -1)*exp(-1)/GAMMA(n-4)):
seq(a(n), n = 5 .. 24); # Miles Wilson, Aug 04 2024

Mathematica

With[{nn=30},Drop[CoefficientList[Series[Exp[-x]/(1-x) x^5/5!,{x,0,nn}],x]Range[0,nn]!,5]] (* Harvey P. Dale, Jan 22 2013 *)

PARI

my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-x)*(x^5/5!))) \\ Joerg Arndt, Feb 17 2014

Python

from sympy import binomial
A129135_list, m, x = [], 1, 0
for n in range(5,21):
    x, m = x*n + m*binomial(n,5), -m
    A129135_list.append(x) # Chai Wah Wu, Nov 01 2014

Formula

a(n) = A008290(n,5).

E.g.f.: exp(-x)/(1-x)*(x^5/5!). - Zerinvary Lajos, Apr 03 2009

a(n) = n*a(n-1) - (-1^n)*binomial(n,5) with a(n) = 0 for n = 0,1,2,3,4. - Chai Wah Wu, Nov 01 2014

D-finite with recurrence (-n+5)*a(n) +n*(n-6)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015

O.g.f.: (1/5!)*Sum_{k>=5} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Extensions

Offset corrected by Susanne Wienand, Feb 17 2014