A065164 Permutation t->t+1 of Z, folded to N.

2, 4, 1, 6, 3, 8, 5, 10, 7, 12, 9, 14, 11, 16, 13, 18, 15, 20, 17, 22, 19, 24, 21, 26, 23, 28, 25, 30, 27, 32, 29, 34, 31, 36, 33, 38, 35, 40, 37, 42, 39, 44, 41, 46, 43, 48, 45, 50, 47, 52, 49, 54, 51, 56, 53, 58, 55, 60, 57, 62, 59, 64, 61, 66, 63, 68, 65, 70, 67, 72, 69, 74

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn
• easy

Comments

Corresponds to simple periodic asynchronic site swap pattern ...111111... (tossing one ball from hand to hand forever).

This permutation consists of a single infinite cycle.

This is, starting at a(2) = 4, the same as the "increasing oscillating sequence" shown in Proposition 3.1, p.7 and plotted in the right of figure 1, of Vatter. The same paper, p.4, cites Comtet and uses without giving the A-number of A003319. Abstract: We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley-Wilf limit) at least lambda = approx 2.48187, the unique real root of x^5-2x^4-2x^2-2x-1, thereby establishing a conjecture of Albert and Linton. - Jonathan Vos Post, Jul 18 2008

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 819.

Links

• Michael H. Albert, Robert Brignall, and Vincent Vatter, Large infinite antichains of permutations, arXiv:1212.3346 [math.CO], 2012.
• Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507-519.
• Jay Pantone and Vincent Vatter, Growth rates of permutation classes: categorization up to the uncountability threshold, arXiv:1605.04289 [math.CO], 2016-2019.
• Vincent Vatter, Permutation classes of every growth rate above 2.48188, arXiv:0807.2815 [math.CO], 2008-2009.
• Vincent Vatter, Permutation classes, arXiv:1409.5159 [math.CO], 2014-2015.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Row 1 of A065167. Obtained by composing permutations A014681 and A065190. Inverse permutation: A065168.

Programs

Maple

ss1 := [seq(PerSS(n,1), n=1..120)]; PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);

Mathematica

Join[{2}, LinearRecurrence[{1, 1, -1}, {4, 1, 6}, 100]] (* Amiram Eldar, Aug 08 2023 *)

Formula

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)+1).

a(n) = n + 2*(-1^n) for n > 1. - Frank Ellermann, Feb 12 2002

a(n) = 2*n-a(n-1)-1, n>2. - Vincenzo Librandi, Dec 07 2010, corrected by R. J. Mathar, Dec 07 2010

From Colin Barker, Feb 18 2013: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.

G.f.: x*(3*x^3-5*x^2+2*x+2) / ((x-1)^2*(x+1)). (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 1. - Amiram Eldar, Aug 08 2023