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A070947 Number of permutations on n letters that have only cycles of length 6 or less. 4
1, 1, 2, 6, 24, 120, 720, 4320, 29520, 225360, 1890720, 17169120, 166112640, 1680462720, 18189031680, 209008512000, 2532028896000, 32143053484800, 425585741760000, 5865854258188800, 84489178710067200, 1266667808011315200, 19700712491727974400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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).

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 *)

PROG

(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(map(a, range(31))) # Indranil Ghosh, Dec 29 2017, after Alois P. Heinz

CROSSREFS

Cf. A057693.

Sequence in context: A179352 A179358 A179365 * A267386 A215718 A060727

Adjacent sequences:  A070944 A070945 A070946 * A070948 A070949 A070950

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified November 18 09:46 EST 2019. Contains 329261 sequences. (Running on oeis4.)