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A000449
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Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.
(Formerly M4700 N2009)
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20
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1, 0, 10, 40, 315, 2464, 22260, 222480, 2447445, 29369120, 381798846, 5345183480, 80177752655, 1282844041920, 21808348713320, 392550276838944, 7458455259940905, 149169105198816960, 3132551209175157490, 68916126601853463240
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OFFSET
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3,3
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 3..100
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
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FORMULA
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a(n) = Sum_{j=2..n-3} (-1)^j*n!/(3!*j!).
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
From Paul Weisenhorn, May 30 2010: (Start)
a(n) = binomial(n,3)*A000166(n-3) with 3 fixed-points;
a(n) = binomial(n,k)*A000166(n-k) with k fixed-points.
(End)
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
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MAPLE
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# 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
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MATHEMATICA
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Table[Subfactorial[n - 3]*Binomial[n, 3], {n, 3, 22}] (* Zerinvary Lajos, Jul 10 2009 *)
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PROG
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(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
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CROSSREFS
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Cf. A000166, A000240, A000387, A000475, A008290, A129135.
A diagonal of A008291.
Cf. A170942.
Sequence in context: A060580 A118266 A054885 * A027274 A253674 A016082
Adjacent sequences: A000446 A000447 A000448 * A000450 A000451 A000452
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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