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%I #15 Sep 08 2022 08:45:48
%S 1,2,6,22,86,335,1271,4680,16766,58656,201106,677767,2251011,7382992,
%T 23955716,77010180,245577076,777648145,2447486221,7661760386,
%U 23872087936,74071120682,228988898916,705618033237,2168073549821,6644571015750,20317533778906
%N Number of permutations of length n which avoid the patterns 4231 and 2143.
%D Kremer, Darla; and Shiu, Wai Chee; Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math. 268 (2003), no. 1-3, 171-183. MR1983276 (2004b:05006). See Table 1.
%H Colin Barker, <a href="/A165527/b165527.txt">Table of n, a(n) for n = 1..1000</a>
%H M. H. Albert, M. D. Atkinson, and R. Brignall, <a href="http://users.mct.open.ac.uk/rb8599/papers/2143-4231-enum.pdf">The enumeration of permutations avoiding 2143 and 4231</a>
%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>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (13,-71,213,-386,438,-311,133,-31,3).
%F G.f.: (x-11*x^2+51*x^3-127*x^4+186*x^5-165*x^6+87*x^7-23*x^8+3*x^9) / ((1-3*x)*(1-x)^4*(1-3*x+x^2)^2). - _Vincent Vatter_, Jun 21 2011
%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t CoefficientList[Series[(x-11*x^2+51*x^3-127*x^4+186*x^5-165*x^6+87*x^7 -23*x^8+3*x^9)/((1-3*x)*(1-x)^4*(1-3*x+x^2)^2), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 22 2018 *)
%o (PARI) Vec(x*(1 - 11*x + 51*x^2 - 127*x^3 + 186*x^4 - 165*x^5 + 87*x^6 - 23*x^7 + 3*x^8) / ((1 - x)^4*(1 - 3*x)*(1 - 3*x + x^2)^2) + O(x^30)) \\ _Colin Barker_, Oct 31 2017
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((x -11*x^2+51*x^3-127*x^4+186*x^5-165*x^6+87*x^7 -23*x^8+3*x^9)/((1-3*x)* (1-x)^4*(1-3*x+x^2)^2))); // _G. C. Greubel_, Oct 22 2018
%K nonn,easy
%O 1,2
%A _Vincent Vatter_, Sep 21 2009
%E More terms from _Vincent Vatter_, Jun 21 2011
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