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A000387 Rencontres numbers: number of permutations of [n] with exactly two fixed points.
(Formerly M4138 N1716)
24
0, 0, 1, 0, 6, 20, 135, 924, 7420, 66744, 667485, 7342280, 88107426, 1145396460, 16035550531, 240533257860, 3848532125880, 65425046139824, 1177650830516985, 22375365779822544, 447507315596451070, 9397653627525472260, 206748379805560389951 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Also: odd permutations of length n with no fixed points. - Martin Wohlgemuth (mail(AT)matroid.com), May 31 2003
Also number of cycles of length 2 in all derangements of [n]. Example: a(4)=6 because in the derangements of [4], namely (1432), (1342), (13)(24), (1423), (12)(34), (1243), (1234), (1324), and (14)(23), we have altogether 6 cycles of length 2. - Emeric Deutsch, Mar 31 2009
REFERENCES
A. Kaufmann, Introduction à la combinatorique en vue des applications, Dunod, Paris, 1968 (see p. 92).
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
Chai Wah Wu, Table of n, a(n) for n = 0..200 (first 100 terms from T. D. Noe)
Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
G. Gordon and E. McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.
Piotr Miska, Arithmetic Properties of the Sequence of Derangements and its Generalizations, arXiv:1508.01987 [math.NT], 2015. (see Chapter 5 p. 44)
J. M. Thomas, The number of even and odd absolute permutations of n letters, Bull. Amer. Math. Soc. 31 (1925), 303-304.
M. Wohlgemuth, Derangements revisited
FORMULA
a(n) = Sum_{j=2..n-2} (-1)^j*n!/(2!*j!) = A008290(n,2).
a(n) = (n!/2) * Sum_{i=0..n-2} ((-1)^i)/i!.
a(n) = A000166(n) - A003221(n).
a(n) = A000166(n-2)*binomial(n, 2). - David Wasserman, Aug 13 2004
E.g.f.: z^2*exp(-z)/(2*(1-z)). - Emeric Deutsch, Jul 22 2009
a(n) ~ n!*exp(-1)/2. - Steven Finch, Mar 11 2022
a(n) = n*a(n-1) + (-1^n)*n*(n-1)/2, a(0) = 0. - Chai Wah Wu, Sep 23 2014
a(n) = A003221(n) + (-1)^n*(n-1) (see Miska). - Michel Marcus, Aug 11 2015
O.g.f.: (1/2)*Sum_{k>=2} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017
D-finite with recurrence +(-n+2)*a(n) +n*(n-3)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023
EXAMPLE
a(4)=6 because we have 1243, 1432, 1324, 4231, 3214, and 2134. - Emeric Deutsch, Mar 31 2009
MAPLE
A000387:= n-> -add((n-1)!*add((-1)^k/(k-1)!, j=0..n-1), k=1..n-1)/2: seq(A000387(n), n=0..25); # Zerinvary Lajos, May 18 2007
A000387 := n -> (-1)^n*(hypergeom([-n, 1], [], 1)+n-1)/2:
seq(simplify(A000387(n)), n=0..22); # Peter Luschny, May 09 2017
MATHEMATICA
Table[Subfactorial[n - 2]*Binomial[n, 2], {n, 0, 22}] (* Zerinvary Lajos, Jul 10 2009 *)
PROG
(Python)
A145221_list, m, x = [], 1, 0
for n in range(201):
x, m = x*n + m*(n*(n-1)//2), -m
A145221_list.append(x) # Chai Wah Wu, Sep 23 2014
(PARI) my(x='x+O('x^33)); concat([0, 0], Vec( serlaplace(exp(-x)/(1-x)*(x^2/2!)) ) ) \\ Joerg Arndt, Feb 19 2014
(PARI) a(n) = ( n!*sum(r=2, n, (-1)^r/r!) - (-1)^(n-1)*(n-1))/2; \\ Michel Marcus, Apr 22 2016
CROSSREFS
Column k=2 of A008290.
Cf. A003221.
A diagonal of A008291.
Cf. A170942.
Sequence in context: A333896 A114959 A000386 * A145221 A332391 A027148
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Prepended a(0)=a(1)=0, Joerg Arndt, Apr 22 2016
STATUS
approved

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