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A001189 Number of degree-n permutations of order exactly 2.
(Formerly M2801 N1127)
39
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
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.
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)
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 *)
PROG
(PARI) {a(n) = sum(j=1, floor(n/2), n!/(j!*(n-2*j)!*2^j))}; \\ G. C. Greubel, May 14 2019
(Magma) m:=30; R<x>:=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
(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
CROSSREFS
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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)