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A000387 Rencontres numbers: number of permutations of [n] with exactly two fixed points.
(Formerly M4138 N1716)
27
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 a 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((-1)^j*n!/(2!*j!), j=2..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) = A145221(n). - Mark van Hoeij, Jul 06 2010

a(n) ~ n!*exp(-3/2)/2. - Geoffrey Critzer, Nov 17 2013

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

EXAMPLE

a(4)=6 because we have 1243, 1432, 1324, 4231, 3214, and 2134. - Emeric Deutsch, Mar 31 2009

MAPLE

a:= n-> -add((n-1)!*add((-1)^k/(k-1)!, j=0..n-1), k=1..n-1)/2: seq(a(n), n=0..25); # Zerinvary Lajos, May 18 2007

MATHEMATICA

Table[Subfactorial[n - 2]*Binomial[n, 2], {n, 0, 22}] (* Zerinvary Lajos, Jul 10 2009 *)

PROG

(Python)

from __future__ import division

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) 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

Cf. A008290, A000166, A000240, A000449, A000475, A129135, A003221.

A diagonal of A008291.

Cf. A170942.

Sequence in context: A074013 A114959 A000386 * A145221 A027148 A095854

Adjacent sequences:  A000384 A000385 A000386 * A000388 A000389 A000390

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Prepended a(0)=a(1)=0, Joerg Arndt, Apr 22 2016

STATUS

approved

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Last modified March 26 23:19 EDT 2017. Contains 284141 sequences.