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%I #29 Jul 07 2024 06:35:29
%S 1,1,2,6,22,87,352,1428,5768,23156,92416,367007,1451780,5725959,
%T 22535868,88566290,347742688,1364637732,5353992916,21005649217,
%U 82425637860,323523434437,1270281675368,4989615315114,19607400037358,77084254889327,303184014866196,1193001145648675
%N Number of permutations of length n which avoid the patterns 4231 and 3214.
%H G. C. Greubel, <a href="/A165532/b165532.txt">Table of n, a(n) for n = 0..1000</a>
%H Darla Kremer and Wai Chee Shiu, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00042-6">Finite transition matrices for permutations avoiding pairs of length four patterns</a>, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
%H Sam Miner, <a href="https://arxiv.org/abs/1610.01908">Enumeration of several two-by-four classes</a>, arXiv:1610.01908 [math.CO], 2016.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Enumerations_of_specific_permutation_classes#Classes_avoiding_two_patterns_of_length_4">Permutation classes avoiding two patterns of length 4</a>.
%F G.f.: (2 - 13*x + 26*x^2 - 17*x^3 + 4*x^4 - x*(1 - 2*x - x^2)*sqrt(1 - 4*x))/(2*sqrt(1 - 4*x)*(1 - 3*x + x^2)^2). - _G. C. Greubel_, Oct 22 2018
%F a(n) ~ 2^(2*n + 4) / (25*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 07 2024
%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t CoefficientList[Series[(2 - 13*x + 26*x^2 - 17*x^3 + 4*x^4 - x*(1 - 2*x - x^2)*Sqrt[1 - 4*x])/(2*Sqrt[1 - 4*x]*(1 - 3*x + x^2)^2), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 22 2018 *)
%o (PARI) x='x+O('x^50); Vec((2 - 13*x + 26*x^2 - 17*x^3 + 4*x^4 - x*(1 - 2*x - x^2)*sqrt(1 - 4*x))/(2*sqrt(1 - 4*x)*(1 - 3*x + x^2)^2)) \\ _G. C. Greubel_, Oct 22 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((2 - 13*x + 26*x^2 - 17*x^3 + 4*x^4 - x*(1 - 2*x - x^2)*Sqrt(1 - 4*x))/( 2*Sqrt(1 - 4*x)*(1 - 3*x + x^2)^2))); // _G. C. Greubel_, Oct 22 2018
%K nonn
%O 0,3
%A _Vincent Vatter_, Sep 21 2009
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 09 2015
%E a(13)-a(15) from _Lars Blomberg_, Apr 26 2018
%E Terms a(16) onward added by _G. C. Greubel_, Oct 22 2018