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

0,3

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.

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)))). - Vladimir Kruchinin, Jan 26 2011

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 *)

Crossrefs

Cf. A057693.

Sequence in context: A179351 A179357 A179364 * A267385 A060726 A152332

Adjacent sequences: A070943 A070944 A070945 * A070947 A070948 A070949

Keyword

nonn

Author

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

Status

approved