A070947 Number of permutations on n letters that have only cycles of length 6 or less.

1, 1, 2, 6, 24, 120, 720, 4320, 29520, 225360, 1890720, 17169120, 166112640, 1680462720, 18189031680, 209008512000, 2532028896000, 32143053484800, 425585741760000, 5865854258188800, 84489178710067200, 1266667808011315200, 19700712491727974400

Offset: 0

Author

N. J. A. Sloane and Sharon Sela, May 18 2002

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..514
• P. L. Krapivsky, J. M. Luck, Coverage fluctuations in theater models, arXiv:1902.04365 [cond-mat.stat-mech], 2019.
• R. Petuchovas, Asymptotic analysis of the cyclic structure of permutations, arXiv:1611.02934 [math.CO], p. 6, 2016.

Crossrefs

Cf. A057693.

Programs

Maple

with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(6):seq(count(A, size=n), n=0..21); # Zerinvary Lajos, Jun 11 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*(j-1)!, j=1..min(n, 6)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 28 2017

Mathematica

terms = 22; CoefficientList[Exp[-Log[1-x] + O[x]^7 // Normal] + O[x]^terms, x]*Range[0, terms-1]! (* Jean-François Alcover, Dec 28 2017 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial, factorial as f
@cacheit
def a(n): return 1 if n==0 else sum(a(n-j)*binomial(n - 1, j - 1)*f(j - 1) for j in range(1, min(n, 6)+1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Dec 29 2017, after Alois P. Heinz

Formula

E.g.f.: exp(x+1/2*x^2+1/3*x^3+1/4*x^4+1/5*x^5+1/6*x^6).