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A363433 Number of (123,231)-avoiding stabilized-interval-free permutations of size n.

%I A363433 #19 Nov 18 2023 18:08:54
%S A363433 1,1,1,1,1,2,3,3,5,5,7,7,10,9,13,12,16,15,20,18,24,22,28,26,33,30,38,
%T A363433 35,43,40,49,45,55,51,61,57,68,63,75,70,82,77,90,84,98,92,106,100,115,
%U A363433 108,124,117,133,126,143,135,153,145,163,155,174,165,185,176,196
%N A363433 Number of (123,231)-avoiding stabilized-interval-free permutations of size n.
%C A363433 A stabilized-interval-free (SIF) permutation on [n] = {1, 2, ..., n} is one that does not stabilize any proper subinterval of [n].
%H A363433 Paolo Xausa, <a href="/A363433/b363433.txt">Table of n, a(n) for n = 0..10000</a>
%H A363433 Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, <a href="https://arxiv.org/abs/2306.03155">Pattern-avoiding stabilized-interval-free permutations</a>, arXiv:2306.03155 [math.CO], 2023.
%H A363433 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,-1,-2,0,1).
%F A363433 G.f.: (x^9 + x^8 - 3*x^6 - 2*x^5 + x^4 + 2*x^3 + x^2 - x - 1)/((x^2 + x + 1)*(x + 1)^2*(x - 1)^3).
%F A363433 E.g.f.: (144 + 36*x*(2 + x) + (3*x^2 + 15*x - 80)*cosh(x) + 8*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (3*x^2 + 33*x + 1)*sinh(x))/72. - _Stefano Spezia_, Jul 01 2023
%F A363433 144*a(n) = 16*A131713(n) +42*n -79 +6*n^2 -81*(-1)^n +18*n*(-1)^n , for n>=3. - _R. J. Mathar_, Jul 17 2023
%e A363433 For n from 1 to 5 the six permutations (1+1+1+1+2) are 1, 21, 312, 4312, 54132, 54213.
%p A363433 A131713 := proc(n)
%p A363433     op(1+modp(n,3),[1,-2,1]) ;
%p A363433 end proc:
%p A363433 A363433 := proc(n)
%p A363433     if n < 3 then
%p A363433         1;
%p A363433     else
%p A363433         16*A131713(n) +42*n-79+6*n^2-81*(-1)^n+18*n*(-1)^n;
%p A363433         %/144 ;
%p A363433     end if;
%p A363433 end proc:
%p A363433 seq(A363433(n),n=0..20) ; # _R. J. Mathar_, Jul 17 2023
%t A363433 LinearRecurrence[{0,2,1,-1,-2,0,1},{1,1,1,1,1,2,3,3,5,5},100] (* _Paolo Xausa_, Nov 18 2023 *)
%o A363433 (PARI) Vec((x^9 + x^8 - 3*x^6 - 2*x^5 + x^4 + 2*x^3 + x^2 - x - 1)/((x^2 + x + 1)*(x + 1)^2*(x - 1)^3) + O(x^65)) \\ _Michel Marcus_, Jul 01 2023
%Y A363433 Cf. A075834, A363431, A363432.
%K A363433 nonn,easy
%O A363433 0,6
%A A363433 _Juan B. Gil_, Jun 30 2023