cp's OEIS Frontend

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

%I A065164 #47 Aug 08 2023 03:38:21
%S A065164 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,
%T A065164 25,30,27,32,29,34,31,36,33,38,35,40,37,42,39,44,41,46,43,48,45,50,47,
%U A065164 52,49,54,51,56,53,58,55,60,57,62,59,64,61,66,63,68,65,70,67,72,69,74
%N A065164 Permutation t->t+1 of Z, folded to N.
%C A065164 Corresponds to simple periodic asynchronic site swap pattern ...111111... (tossing one ball from hand to hand forever).
%C A065164 This permutation consists of a single infinite cycle.
%C A065164 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
%D A065164 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 819.
%H A065164 Michael H. Albert, Robert Brignall, and Vincent Vatter, <a href="https://arxiv.org/abs/1212.3346">Large infinite antichains of permutations</a>, arXiv:1212.3346 [math.CO], 2012.
%H A065164 Joe Buhler and R. L. Graham, <a href="http://www.cecm.sfu.ca/organics/papers/buhler/index.html">Juggling Drops and Descents</a>, Amer. Math. Monthly, 101, (no. 6) 1994, 507-519.
%H A065164 Jay Pantone and Vincent Vatter, <a href="https://arxiv.org/abs/1605.04289">Growth rates of permutation classes: categorization up to the uncountability threshold</a>, arXiv:1605.04289 [math.CO], 2016-2019.
%H A065164 Vincent Vatter, <a href="https://arxiv.org/abs/0807.2815">Permutation classes of every growth rate above 2.48188</a>, arXiv:0807.2815 [math.CO], 2008-2009.
%H A065164 Vincent Vatter, <a href="https://arxiv.org/abs/1409.5159">Permutation classes</a>, arXiv:1409.5159 [math.CO], 2014-2015.
%H A065164 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%H A065164 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A065164 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).
%F A065164 a(n) = n + 2*(-1^n) for n > 1. - _Frank Ellermann_, Feb 12 2002
%F A065164 a(n) = 2*n-a(n-1)-1, n>2. - _Vincenzo Librandi_, Dec 07 2010, corrected by _R. J. Mathar_, Dec 07 2010
%F A065164 From _Colin Barker_, Feb 18 2013: (Start)
%F A065164 a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
%F A065164 G.f.: x*(3*x^3-5*x^2+2*x+2) / ((x-1)^2*(x+1)). (End)
%F A065164 Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 1. - _Amiram Eldar_, Aug 08 2023
%p A065164 ss1 := [seq(PerSS(n,1), n=1..120)]; PerSS := (n,c) -> Z2N(N2Z(n)+c);
%p A065164 N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
%t A065164 Join[{2}, LinearRecurrence[{1, 1, -1}, {4, 1, 6}, 100]] (* _Amiram Eldar_, Aug 08 2023 *)
%Y A065164 Row 1 of A065167. Obtained by composing permutations A014681 and A065190. Inverse permutation: A065168.
%K A065164 nonn,easy
%O A065164 1,1
%A A065164 _Antti Karttunen_, Oct 19 2001