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

0,6

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.

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.

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. *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^5/5) )) \\ G. C. Greubel, May 14 2019
(Magma) m:=30; R<x>:=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
(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

Crossrefs

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

Column k=5 of A008307.

Sequence in context: A017042 A100255 A305269 * A193438 A139152 A123014

Adjacent sequences: A052498 A052499 A052500 * A052502 A052503 A052504

Keyword

nonn,nice,easy

Author

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

Status

approved