A001189 Number of degree-n permutations of order exactly 2.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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:1403.5477 [q-bio.BM], 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
• Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.
• Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., April 2004.

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

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2) -Exp(x) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, May 14 2019

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 *)

PARI

{a(n) = sum(j=1,floor(n/2), n!/(j!*(n-2*j)!*2^j))}; \\ G. C. Greubel, May 14 2019

Sage

m = 30; T = taylor(exp(x +x^2/2) - exp(x), x, 0, m); a=[factorial(n)*T.coefficient(x, n) for n in (0..m)]; a[1:] # G. C. Greubel, May 14 2019

Formula

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

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.

From Henry Bottomley, May 03 2001: (Start)

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

a(n) = Sum_{j=1..floor(n/2)} n!/(j!*(n-2*j)!*(2^j)). (End)