A088436 Number of permutations in the symmetric group S_n that have exactly one transposition in their cycle decomposition.

0, 1, 3, 6, 30, 225, 1575, 12180, 109620, 1100925, 12110175, 145259730, 1888376490, 26438216805, 396573252075, 6345155817000, 107867648889000, 1941617990136825, 36890741812599675, 737814829704702750

Offset: 1

Author

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

Keywords

• nonn

Examples

			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)
		

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

Crossrefs

Cf. A000266, A027616, A000240.

Programs

Magma

m:=32; R:=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

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 *)

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

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

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

Extensions

More terms from Wolfdieter Lang, Feb 22 2008