A003483 Number of square permutations of n elements.

1, 1, 1, 3, 12, 60, 270, 1890, 14280, 128520, 1096200, 12058200, 139043520, 1807565760, 22642139520, 339632092800, 5237183952000, 89032127184000, 1475427973219200, 28033131491164800, 543494606861606400, 11413386744093734400, 235075995738558374400, 5406747901986842611200, 126214560713084056012800

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

"A permutation P has a square root if and only if the numbers of cycles of P that have each even length are even numbers." [Theorem 4.8.1. on p.147 from the Wilf reference]. - Joerg Arndt, Sep 08 2014

Examples

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

References

• 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.
• H. S. Wilf, Generatingfunctionology, 3rd ed., A K Peters Ltd., Wellesley, MA, 2006, p. 157.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 101 terms from N. J. A. Sloane)
• Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
• Joseph Blum, Letters to N. J. A. Sloane, 1974
• J. Blum, Enumeration of the square permutations in S_n, J. Combin. Theory, A 17 (1974), 156-161.
• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 36.
• Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
• P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics, arXiv:math.CO/0606370, p. 18, Proposition 2.
• Yuewen Luo, Counting Permutations in S_{2n} and S_{2n+1}, arXiv:2407.07366 [math.CO], 2024.
• M. R. Pournaki, On the number of even permutations with roots, The Australasian Journal of Combinatorics, Volume 45, 2009, pp. 37-42.
• N. Pouyanne, On the number of permutations admitting an m-th root, Electron. J. Combin., 9 (2002), #R3.
• Bob Smith and N. J. A. Sloane, Correspondence, 1979
• H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148, Eq. 4.8.1.

Crossrefs

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

Column k=2 of A247005.

Cf. A246945, A247621.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i, 2)=0 and irem(j, 2)=1, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2014

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-François Alcover, Oct 05 2011, after g.f. *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 1, 0, (i-1)!^j* multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Sqrt[ (1 + x) / (1 - x)] Product[ Cosh[ x^k / k], {k, 2, n, 2}], {x, 0, n}]]; (* Michael Somos, Jul 11 2018 *)

PARI

N=66; x='x+O('x^66);
Vec(serlaplace( sqrt((1+x)/(1-x))*prod(k=1,N, cosh(x^(2*k)/(2*k)))))
\\ Joerg Arndt, Sep 08 2014

Formula

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

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

Asymptotics: a(n) ~ n! * sqrt(2/(n*Pi)) * e^G, where e^G = Product_{k>=1} cosh(1/(2k)) ~ 1.22177951519253683396485298445636121278881... (see A246945). - corrected by Vaclav Kotesovec, Sep 13 2014

G = Sum_{j>=1} (-1)^(j + 1) * Zeta(2*j)^2 * (1 - 1/2^(2*j)) / (j * Pi^(2*j)). - Vaclav Kotesovec, Sep 20 2014

Extensions

More terms from Vladeta Jovovic, Mar 28 2001

Additional comments from Michael Somos, Jun 27 2002

Minor edits by Vaclav Kotesovec, Sep 16 2014 and Sep 21 2014