A052501 Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5.

1, 1, 1, 1, 1, 25, 145, 505, 1345, 3025, 78625, 809425, 4809025, 20787625, 72696625, 1961583625, 28478346625, 238536558625, 1425925698625, 6764765838625, 189239120970625, 3500701266525625, 37764092547420625, 288099608198025625

Offset: 0

Author

N. J. A. Sloane, Jan 15 2000; encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• nonn
• nice
• easy

Comments

The number of degree-n permutations of order exactly p (where p is prime) satisfies a(n) = a(n-1) + (1 + a(n-p))*(n-1)!/(n-p)! with a(n)=0 if p>n. Also a(n) = Sum_{j=1..floor(n/p)} (n!/(j!*(n-p*j)!*(p^j))).

These are the telephone numbers T^(5)n of [Artioli et al., p. 7]. - _Eric M. Schmidt, Oct 12 2017

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
• Tomislav Došlic, Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). - _N. J. A. Sloane_, May 01 2012
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 26
• M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6.
• L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=5 of A008307.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^5/5) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019

Maple

spec := [S,{S=Set(Union(Cycle(Z,card=1),Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

max = 30; CoefficientList[ Series[ Exp[x + x^5/5], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Feb 15 2012, after e.g.f. *)

PARI

my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^5/5) )) \\ G. C. Greubel, May 14 2019

Sage

m = 30; T = taylor(exp(x + x^5/5), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

Formula

E.g.f.: exp(x + x^5/5).

a(n+5) = a(n+4) + (24 +50*n +35*n^2 +10*n^3 +n^4)*a(n), with a(0)= ... = a(4) = 1.

a(n) = a(n-1) + a(n-5)*(n-1)!/(n-5)!.

a(n) = Sum_{j = 0..floor(n/5)} n!/(5^j * j! * (n-5*j)!).

a(n) = A059593(n) + 1.