

A000246


Number of permutations in the symmetric group S_n that have odd order.
(Formerly M2824 N1137)


44



1, 1, 1, 3, 9, 45, 225, 1575, 11025, 99225, 893025, 9823275, 108056025, 1404728325, 18261468225, 273922023375, 4108830350625, 69850115960625, 1187451971330625, 22561587455281875, 428670161650355625, 9002073394657468125, 189043541287806830625
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OFFSET

0,4


COMMENTS

Michael Reid (mreid(AT)math.umass.edu) points out that the e.g.f. for the number of permutations of odd order can be obtained from the cycle index for S_n, F(Y; X1, X2, X3, ... ) := e^(X1 Y + X2 Y^2/2 + X3 Y^3/3 + ... ) and is F(Y, 1, 0, 1, 0, 1, 0, ... ) = sqrt((1 + Y)/(1  Y)).
a(n) appears to be the number of permutations on [n] whose updown signature has nonnegative partial sums. For example, the updown signature of (2,4,5,1,3) is (+1,+1,1,+1) with nonnegative partial sums 1,2,1,2 and a(3)=3 counts (1,2,3), (1,3,2), (2,3,1).  David Callan, Jul 14 2006
This conjecture has been confirmed, see Bernardi, Duplantier, Nadeau link.
a(n) is the number of permutations of [n] for which all lefttoright minima occur in odd locations in the permutation. For example, a(3)=3 counts 123, 132, 231. Proof: For such a permutation of length 2n, you can append 1,2,..., or 2n+1 (2n+1 choices) and increase by 1 the original entries that weakly exceed the appended entry. This gives all such permutations of length 2n+1. But if the original length is 2n1, you cannot append 1 (for then 1 would be a lefttoright min in an even location) so you can only append 2,3,..., or 2n (2n1 choices). This count matches the given recurrence relation a(2n)=(2n1)a(2n1), a(2n+1)=(2n+1)a(2n).  David Callan, Jul 22 2008
a(n) is the nth derivative of exp(arctanh(x)) at x = 0.  Michel Lagneau, May 11 2010
a(n) is the absolute value of the Moebius number of the odd partition poset on a set of n+1 points, where the odd partition poset is defined to be the subposet of the partition poset consisting of only partitions using odd part size (as well as the maximum element for n even).  Kenneth M Monks, May 06 2012
Number of permutations in S_n in which all cycles have odd length.  Michael Somos, Mar 17 2019


REFERENCES

H.D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 87.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
Joel Barnes, Conformal welding of uniform random trees, Ph. D. Dissertation, Univ. Washington, 2014.
Olivier Bernardi, Bertrand Duplantier and Philippe Nadeau, A Bijection Between WellLabelled Positive Paths and Matchings, Séminaire Lotharingien de Combinatoire (2010), volume 63, Article B63e.
A. Edelman and M. La Croix, The Singular Values of the GUE (Less is More), arXiv preprint arXiv:1410.7065 [math.PR], 20142015. See Table 1.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
A. Ghitza and A. McAndrew, Experimental evidence for Maeda's conjecture on modular forms, arXiv preprint arXiv:1207.3480 [math.NT], 2012.
Y. Cha, Closed form solutions of difference equations (2011) PhD Thesis, Florida State University, page 24
Dmitry Kruchinin, Integer properties of a composition of exponential generating functions, arXiv:1211.2100 [math.NT], 2012.
Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.
Kenneth M. Monks, An Elementary Proof of the Explicit Formula for the Möbius Number of the Odd Partition Poset, J. Int. Seq., Vol. 21 (2018), Article 18.9.6.
Qingchun Ren, Ordered Partitions and Drawings of Rooted Plane Trees, arXiv preprint arXiv:1301.6327 [math.CO], 2013. See Lemma 15.
Marko Riedel, et al. From combinatorial class to recurrence to closed form, Mathematics Stack Exchange.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729734 and 113 (2006), 670.
Sam Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.
Allen Wang, Permutations with UpDown Signatures of Nonnegative Partial Sums, MIT PRIMES Conference (2018).
David G. L. Wang and T. Zhao, The peak and descent statistics over ballot permutations, arXiv:2009.05973 [math.CO], 2020.
Index entries for sequences related to groups


FORMULA

E.g.f.: sqrt(1x^2)/(1x) = sqrt((1+x)/(1x)).
a(2*k) = (2*k1)*a(2*k1), a(2*k+1) = (2*k+1)*a(2*k), for k >= 0, with a(0) = 1.
Let b(1)=0, b(2)=1, b(k+2)=b(k+1)/k + b(k); then a(n+1) = n!*b(n+2).  Benoit Cloitre, Sep 03 2002
a(n) = Sum_{k=0..floor((n1)/2)} (2k)! * C(n1, 2k) * a(n2k1) for n > 0.  Noam Katz (noamkj(AT)hotmail.com), Feb 27 2001
Also successive denominators of Wallis's approximation to Pi/2 (unreduced): 1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * .., for n >= 1.
Dfinite with recurrence: a(n) = a(n1) + (n1)*(n2)*a(n2).  Benoit Cloitre, Aug 30 2003
a(n) is asymptotic to (n1)!*sqrt(2*n/Pi).  Benoit Cloitre, Jan 19 2004
a(n) = n! * binomial(n1, floor((n1)/2)) / 2^(n1), n > 0.  Ralf Stephan, Mar 22 2004
E.g.f.: e^atanh(x), a(n) = n!*Sum_{m=1..n} Sum_{k=m..n} 2^(km)*Stirling1(k,m)*binomial(n1,k1)/k!, n > 0, a(0)=1.  Vladimir Kruchinin, Dec 12 2011
G.f.: G(0) where G(k) = 1 + x*(4*k1)/((2*k+1)*(x1)  x*(x1)*(2*k+1)*(4*k+1)/(x*(4*k+1) + 2*(x1)*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3step).  Sergei N. Gladkovskii, Jul 24 2012
G.f.: 1 + x*(G(0)  1)/(x1) where G(k) = 1  (2*k+1)/(1x/(x  1/(1  (2*k+1)/(1x/(x  1/G(k+1) ))))); (continued fraction).  Sergei N. Gladkovskii, Jan 15 2013
G.f.: G(0), where G(k) = 1 + x*(2*k+1)/(1  x*(2*k+1)/(x*(2*k+1) + 1/G(k+1))); (continued fraction).  Sergei N. Gladkovskii, Jun 07 2013
For n >= 1, a(2*n) = (2*n1)!!^2, a(2*n+1) = (2*n+1)*(2*n1)!!^2.  Vladimir Shevelev, Dec 01 2013
E.g.f.: arcsin(x)  sqrt(1x^2) + 1 for a(0) = 0, a(1) = a(2) = a(3) = 1.  G. C. Greubel, May 01 2015
Sum_{n>1} 1/a(n) = (L_0(1) + L_1(1))*Pi/2, where L is the modified Struve function.  Peter McNair, Mar 11 2022


EXAMPLE

For the Wallis numerators, denominators and partial products see A001900.  Wolfdieter Lang, Dec 06 2017


MAPLE

a:= proc(n) option remember; `if`(n<2, 1,
a(n1) +(n1)*(n2)*a(n2))
end:
seq(a(n), n=0..25); # Alois P. Heinz, May 14 2018


MATHEMATICA

a[n_] := a[n] = a[n1]*(n+Mod[n, 2]1); a[0] = 1; Table[a[n], {n, 0, 20}] (* JeanFrançois Alcover, Nov 21 2011, after Pari *)
a[n_] := a[n] = (n2)*(n3)*a[n2] + a[n1]; a[0] := 0; a[1] := 1; Table[a[i], {i, 0, 20}] (* or *) RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(n2)*(n3)a[n2]+a[n1]}, a, {n, 20}] (* G. C. Greubel, May 01 2015 *)
CoefficientList[Series[Sqrt[(1+x)/(1x)], {x, 0, 50}], x]*Table[k!, {k, 0, 20}] (* Stefano Spezia, Oct 07 2018 *)


PROG

(PARI) a(n)=if(n<1, !n, a(n1)*(n+n%21))
(PARI) Vec( serlaplace( sqrt( (1+x)/(1x) + O(x^55) ) ) )
(PARI) a(n)=prod(k=3, n, k+k%21) \\ Charles R Greathouse IV, May 01 2015
(PARI) a(n)=(n!/(n\2)!>>(n\2))^2/if(n%2, n, 1) \\ Charles R Greathouse IV, May 01 2015
(Haskell)
a000246 n = a000246_list !! n
a000246_list = 1 : 1 : zipWith (+)
(tail a000246_list) (zipWith (*) a000246_list a002378_list)
 Reinhard Zumkeller, Feb 27 2012
(Magma) I:=[1, 1]; [n le 2 select I[n] else Self(n1)+(n^25*n+6)*Self(n2): n in [1..30]]; // Vincenzo Librandi, May 02 2015


CROSSREFS

Cf. A001900, A059838, A002867.
Bisections are A001818 and A079484.
Row sums of unsigned triangle A049218 and of A111594, A262125.
Main diagonal of A262124.
Cf. A002019.
Sequence in context: A262133 A262134 A262135 * A247006 A103620 A138315
Adjacent sequences: A000243 A000244 A000245 * A000247 A000248 A000249


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



