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A088436 Number of permutations in the symmetric group S_n that have exactly one transposition in their cycle decomposition. 4
0, 1, 3, 6, 30, 225, 1575, 12180, 109620, 1100925, 12110175, 145259730, 1888376490, 26438216805, 396573252075, 6345155817000, 107867648889000, 1941617990136825, 36890741812599675, 737814829704702750 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 189, Exercise 19 for k=1. With (-1)^k omitted.

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..440

FORMULA

a(n) = (n!/2)*Sum_{j=0..floor(n/2)-1} (-1)^j/(j!*2^j), n >= 1.

E.g.f.: x^2/(1-x)/2*exp(-x^2/2). - Vladeta Jovovic, Nov 09 2003

From Paul Weisenhorn, Jun 02 2010: (Start)

In general, for k cycles of length 2,

a(n) = n!*Sum_{j=k..floor(n/2)} (-1)^j/((j-k)!*2^j*k!).

G.f.: (exp(-z^2/2)*z^2*k)/((1-z)*2^k*k!). (End)

a(n) ~ exp(-1/2)/2 * n!. - Vaclav Kotesovec, Mar 18 2014

EXAMPLE

From Bernard Schott, Feb 19 2019: (Start)

For S_4, the six permutations that have exactly one transposition in their cycle decomposition are (12)(3)(4), (13)(2)(4), (14)(2)(3), (23)(1)(4), (24)(1)(3), (34)(1)(2).

For S_5, there are exactly 10 transpositions: (12), (13), (14), (15), (23), (24), (25), (34), (35), (45), and for each transposition, there are 3 permutations that have exactly this transposition and no other transposition in their cycle decomposition; for example, for transposition (12), these three permutations: (12)(3)(4)(5), (12)(345), (12)(354), so a(5) = 10 * 3 = 30. (End)

MAPLE

G=(exp(-z^2/2)*z^2*k)/((1-z)*2^k*k!): Gser=series(G, z=0, 21):

for n from 2*k to 20 do a(n)=n!*coeff(Gser, z, n): end do: # Paul Weisenhorn, Jun 02 2010

MATHEMATICA

d=Exp[-x^2/2]/(1-x); Range[0, 20]! CoefficientList[Series[(x^2/2! )d, {x, 0, 20}], x] (* Geoffrey Critzer, Nov 29 2011 *)

PROG

(PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace( x^2*exp(-x^2/2)/(2*(1-x)) ))) \\ G. C. Greubel, Feb 19 2019

(MAGMA) m:=32; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x^2*Exp(-x^2/2)/(2*(1-x)) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Feb 19 2019

(Sage) m = 30; T = taylor(x^2*exp(-x^2/2)/(2*(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (1..m)] # G. C. Greubel, Feb 19 2019

CROSSREFS

Cf. A000266, A027616, A000240.

Sequence in context: A157534 A133799 A262022 * A088506 A061137 A012280

Adjacent sequences:  A088433 A088434 A088435 * A088437 A088438 A088439

KEYWORD

nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003

EXTENSIONS

More terms from Wolfdieter Lang, Feb 22 2008

STATUS

approved

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Last modified April 7 07:50 EDT 2020. Contains 333292 sequences. (Running on oeis4.)