A000449 Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.

1, 0, 10, 40, 315, 2464, 22260, 222480, 2447445, 29369120, 381798846, 5345183480, 80177752655, 1282844041920, 21808348713320, 392550276838944, 7458455259940905, 149169105198816960, 3132551209175157490, 68916126601853463240

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 3..100
• 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.

A diagonal of A008291.

Cf. A170942.

Programs

Maple

# with k fixed-points:
G:=exp(-z)*z^k/((1-z)*k!: Gser:=series(G,z,21):
for n from k to 20 do a(n)=n!*coeff(Gser,z,n): end do: # Paul Weisenhorn, May 30 2010

Mathematica

Table[Subfactorial[n - 3]*Binomial[n, 3], {n, 3, 22}] (* Zerinvary Lajos, Jul 10 2009 *)

PARI

my(x='x+O('x^66)); Vec( serlaplace(exp(-x)/(1-x)*(x^3/3!)) ) \\ Joerg Arndt, Feb 19 2014

Python

A000449_list, m, x = [], 1, 0
for n in range(3,21):
    x, m = x*n + m*(n*(n-1)*(n-2)//6), -m
    A000449_list.append(x) # Chai Wah Wu, Sep 23 2014

Formula

a(n) = Sum_{j=2..n-3} (-1)^j*n!/(3!*j!) = A008290(n,3).

For n >= 3 a(n) = C(n, 3) * A000166(n-3) = 1/6 * n! * Sum_{k=0..n-3} (-1)^k/k!. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 14 2001

E.g.f.: 1/(exp(x)*(1-x))*(x^3)/6. - Wenjin Woan, Nov 20 2008

E.g.f.: x^3*exp(-x)/(3!*(1-x)). - Geoffrey Critzer, Nov 03 2012

a(n) ~ n! * exp(-1)/6. - Vaclav Kotesovec, Mar 17 2014

a(n) = n*a(n-1) - (-1^n)*n*(n-1)*(n-2)/6, a(n) = 0 for n= 0, 1, 2. - Chai Wah Wu, Sep 23 2014

O.g.f.: (1/6)*Sum_{k>=3} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017

D-finite with recurrence (-n+3)*a(n) +n*(n-4)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023