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A003483 Number of square permutations of n elements.
(Formerly M2931)
16
1, 1, 1, 3, 12, 60, 270, 1890, 14280, 128520, 1096200, 12058200, 139043520, 1807565760, 22642139520, 339632092800, 5237183952000, 89032127184000, 1475427973219200, 28033131491164800, 543494606861606400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of permutations p in S_n such that there exists q in S_n with q^2=p.

REFERENCES

Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, 485-515.

J. Blum, Enumeration of the square permutations in S_n, J. Combin. Theory, A 17 (1974), 156-161.

N. Pouyanne, On the number of permutations admitting an m-th root, Electron. J. Combin., 9 (2002), #R3.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.11.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..100

P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics, arXiv:math.CO/0606370

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148, Eq. 4.8.1.

FORMULA

E.g.f.: sqrt((1+x)/(1-x))*Product_{k >= 1} cosh x^(2k)/(2k) [Blum, corrected].

a(2n+1)=(2n+1)a(2n).

Asymptotics: a(n) ~ n! 2/sqrt(n pi) e^G, where e^G = prod_{k>=1} cosh(1/(2k)) ~ 1.22178

EXAMPLE

a(3)=3: permutations with square roots are identity and two 3-cycles.

MATHEMATICA

max = 20; f[x_] := Sqrt[(1 + x)/(1 - x)]*  Product[ Cosh[x^(2*k)/(2*k)], {k, 1, max}]; se = Series[ f[x], {x, 0, max}]; CoefficientList[ se, x]*Range[0, max]! (* Jean-Fran├žois Alcover, Oct 05 2011, after g.f. *)

CROSSREFS

Cf. A103619 (cube root), A103620 (fourth root), A215716 (fifth root), A215717 (sixth root), A215718 (seventh root).

Sequence in context: A127918 A069944 A073996 * A128602 A092803 A181282

Adjacent sequences:  A003480 A003481 A003482 * A003484 A003485 A003486

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vladeta Jovovic, Mar 28 2001

Additional comments from Michael Somos, Jun 27, 2002

STATUS

approved

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Last modified April 16 15:34 EDT 2014. Contains 240600 sequences.