A052504 Number of permutations sigma of [5n] without fixed points such that sigma^5 = Id.

1, 24, 72576, 1743565824, 162193467211776, 41363226782215962624, 23578031983305871878782976, 26242915470187034742010543079424, 51804144968120491069562620291816882176, 168779147605615794796420686413626405734580224, 858246016274098851318874304509764200194078068965376

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(5n) consisting of permutations whose cycle decomposition is a product of n disjoint 5-cycles.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..100
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 29

Crossrefs

Cf. A001147, A052502, A060706, A261317, A261381.

Quintisection of column k=5 of A261430.

Programs

GAP

List([0..15], n-> Factorial(5*n)/(5^n*Factorial(n))) # G. C. Greubel, May 14 2019

Magma

[Factorial(5*n)/(5^n*Factorial(n)): n in [0..15]]; // G. C. Greubel, May 14 2019

Maple

spec := [S,{S=Set(Union(Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

nn = 50; Select[Range[0, nn]! CoefficientList[Series[Exp[x^5/5], {x, 0, nn}], x], # > 0 &]  (* Geoffrey Critzer, Aug 19 2012 *)

PARI

{a(n) = (5*n)!/(5^n*n!)}; \\ G. C. Greubel, May 14 2019

Sage

[factorial(5*n)/(5^n*factorial(n)) for n in (0..15)] # G. C. Greubel, May 14 2019

Formula

a(n) = (5n)! * [x^(5n)] exp(x^5/5).

From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)

a(n) = (5*n)! / (n! * 5^n).

a(0) = 1, a(1) = 24, for n >= 2 a(n) = a(n-1) * C(5*n - 1, 4)* 24 = (5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*a(n-1).

a(n) ~ sqrt(5) * 625^n * (n/e)^(4n). (End)

Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(4*n+1)/(4*n+1)!. Then A'(x)*( 1 - A(x)^4) = 1. Cf. A052502. - Peter Bala, Jan 02 2015