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A070945
Number of permutations on n letters that have only cycles of length 4 or less.
6
1, 1, 2, 6, 24, 96, 456, 2472, 14736, 92304, 632736, 4661856, 36364032, 297668736, 2583425664, 23550535296, 224086162176, 2221083839232, 22976670905856, 246829966447104, 2745834333566976, 31605782067081216, 376290722808502272
OFFSET
0,3
REFERENCES
Dennis P. Walsh, The Number of Permutations with Only Small Cycles, preprint [From Geoffrey Critzer, May 24 2009]
LINKS
P. L. Krapivsky, J. M. Luck, Coverage fluctuations in theater models, arXiv:1902.04365 [cond-mat.stat-mech], 2019.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1 .
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).
MAPLE
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(4):seq(count(A, size=n), n=0..22); # Zerinvary Lajos, Jun 11 2008
G := exp(x+(1/2)*x^2+(1/3)*x^3+(1/4)*x^4): seq(factorial(n)*coeftayl(G, x = 0, n), n = 0 .. 22); # Emeric Deutsch, Jun 21 2009
MATHEMATICA
Table[Sum[Binomial[n, 4 i]*(4 i)!/(i!*4^i)* Sum[Binomial[n - 4 i, 3 j]*(3 j)!/(j!*3^j)* Sum[Binomial[n - 4 i - 3 j, 2 k]*(2 k)!/(k!*2^k), {k, 0, n}], {j, 0, n}], {i, 0, n}], {n, 0, 22}] (* Geoffrey Critzer, May 24 2009 *)
With[{nn = 23, k = 5}, CoefficientList[Exp[-Log[1 - x] + O[x]^k // Normal] + O[x]^nn, x] Range[0, nn - 1]!] (* Michael De Vlieger, Mar 29 2019, after Jean-François Alcover at A070947 *)
CROSSREFS
Cf. A057693.
Sequence in context: A147912 A147903 A147895 * A152320 A267384 A152327
KEYWORD
nonn
AUTHOR
N. J. A. Sloane and Sharon Sela, May 18 2002
STATUS
approved