A070946 Number of permutations on n letters that have only cycles of length 5 or less.

1, 1, 2, 6, 24, 120, 600, 3480, 22800, 164880, 1285920, 10516320, 92931840, 877374720, 8762014080, 91819440000, 1005716908800, 11584953158400, 139521689740800, 1748830512960000, 22750446292531200, 306931140411955200, 4296645083802470400, 62213458150660147200

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..532
• P. L. Krapivsky, J. M. Luck, Coverage fluctuations in theater models, arXiv:1902.04365 [cond-mat.stat-mech], 2019.
• Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
• 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(5):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, 5)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 25 2018

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x+x^2/2+x^3/3+x^4/4+ x^5/5], {x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 24 2016 *)

Formula

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

a(n) = n!*Sum_{k=1..n} (1/k!)*Sum_{r=0..k} binomial(k,r)*Sum_{m=0..r} 2^(m-r)*binomial(r,m)*Sum_{j=0..m} binomial(m,j)*binomial(j,n-m-k-j-r)*3^(j-m)*4^(n-r-m-k-2*j)*5^(m+k+j+r-n) for n > 0. - Vladimir Kruchinin, Jan 26 2011