OFFSET
0,4
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
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.
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
EXAMPLE
a(3) = 3: permutations with square roots are identity and two 3-cycles.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
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
STATUS
approved