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Permutation t->t+1 of Z, folded to N.
6

%I #47 Aug 08 2023 03:38:21

%S 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 25,30,27,32,29,34,31,36,33,38,35,40,37,42,39,44,41,46,43,48,45,50,47,

%U 52,49,54,51,56,53,58,55,60,57,62,59,64,61,66,63,68,65,70,67,72,69,74

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

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

%C This permutation consists of a single infinite cycle.

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

%H 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 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 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 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 Vincent Vatter, <a href="https://arxiv.org/abs/1409.5159">Permutation classes</a>, arXiv:1409.5159 [math.CO], 2014-2015.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F 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 a(n) = n + 2*(-1^n) for n > 1. - _Frank Ellermann_, Feb 12 2002

%F a(n) = 2*n-a(n-1)-1, n>2. - _Vincenzo Librandi_, Dec 07 2010, corrected by _R. J. Mathar_, Dec 07 2010

%F From _Colin Barker_, Feb 18 2013: (Start)

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

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

%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 1. - _Amiram Eldar_, Aug 08 2023

%p ss1 := [seq(PerSS(n,1), n=1..120)]; PerSS := (n,c) -> Z2N(N2Z(n)+c);

%p N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);

%t Join[{2}, LinearRecurrence[{1, 1, -1}, {4, 1, 6}, 100]] (* _Amiram Eldar_, Aug 08 2023 *)

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

%K nonn,easy

%O 1,1

%A _Antti Karttunen_, Oct 19 2001