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A001189 Number of degree-n permutations of order exactly 2.
(Formerly M2801 N1127)
40
0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503, 2390479, 10349535, 46206735, 211799311, 997313823, 4809701439, 23758664095, 119952692895, 618884638911, 3257843882623, 17492190577599, 95680443760575, 532985208200575, 3020676745975551 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Number of set partitions of [n] into blocks of size 2 and 1 with at least one block of size 2. - Olivier Gérard, Oct 29 2007

For n>=2, number of standard Young tableaux with height <= n - 1. That is, all tableaux (A000085) but the one with just one column. [Joerg Arndt, Oct 24 2012]

REFERENCES

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

Alois P. Heinz, Table of n, a(n) for n = 1..800

N. Chheda, M. K. Gupta, RNA as a Permutation, arXiv preprint arXiv:1403.5477, 2014.

R. B. Herrera, The number of elements of given period in finite symmetric group, Amer. Math. Monthly 64, 1957, 488-490.

L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

J. Rangel-Mondragon, Selected Themes in Computational Non-Euclidean Geometry: Part 1, The Mathematica Journal 15 (2013); http://www.mathematica-journal.com/data/uploads/2013/07/Rangel-Mondragon_Selected-1.pdf

Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., April 2004.

FORMULA

E.g.f.: -exp(x)+exp(x+1/2*x^2).

a(n) = A000085(n) - 1.

a(n) = b(n, 2), where b(n, d)=Sum_{k=1..n} (n-1)!/(n-k)! * Sum_{l:lcm{k, l}=d} b(n-k, l), b(0, 1)=1 is the number of degree-n permutations of order exactly d.

a(n) = a(n-1)+(1+a(n-2))*(n-1) = Sum_{j = 1 to floor[n/2]}[n!/(j!*(n-2j)!*(2^j))] = A000085(n)-1. - Henry Bottomley, May 03 2001

MAPLE

a:= proc(n) option remember; `if`(n<3, [0$2, 1][n+1],

      a(n-1) +(n-1) *(1+a(n-2)))

    end:

seq(a(n), n=1..30);  # Alois P. Heinz, Oct 24 2012

# alternative:

A001189 := proc(n)

    local a, prs, p, k ;

    a := 0 ;

    for prs from 1 to n/2 do

        p := product(binomial(n-2*k, 2), k=0..prs-1) ;

        a := a+p/prs!;

    end do:

    a;

end proc:

seq(A001189(n), n=1..13) ; # R. J. Mathar, Jan 04 2017

MATHEMATICA

RecurrenceTable[{a[1]==0, a[2]==1, a[n]==a[n-1]+(1+a[n-2])(n-1)}, a[n], {n, 25}] (* Harvey P. Dale, Jul 27 2011 *)

CROSSREFS

Cf. A001470-A001473, A052501, A053496-A053504, A061121-A061128.

Column k=1 of A143911, column k=2 of A080510, A182222. - Alois P. Heinz, Oct 24 2012

Column k=2 of A057731. - Alois P. Heinz, Feb 14 2013

Sequence in context: A183111 A132835 A191354 * A212352 A198180 A101786

Adjacent sequences:  A001186 A001187 A001188 * A001190 A001191 A001192

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.