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%I #31 Sep 08 2022 08:45:48
%S 1,1,2,6,22,86,336,1282,4758,17234,61242,214594,744594,2566594,
%T 8809442,30157826,103082050,352045314,1201795970,4101913602,
%U 13999868162,47782800386,163095158274,556717514754,1900427035650,6487635578882,22148113283074,75613356769282
%N Number of permutations of length n which avoid the patterns 1324 and 4231.
%H G. C. Greubel, <a href="/A165528/b165528.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Albert, M. Atkinson, and V. Vatter, <a href="https://doi.org/10.37236/225">Counting 1324-, 4231-avoiding permutations</a>, The Electronic Journal of Combinatorics, Volume 16, Issue 1 (2009).
%H Christian Bean, <a href="https://hdl.handle.net/20.500.11815/1184">Finding structure in permutation sets</a>, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
%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 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_07">Index entries for linear recurrences with constant coefficients</a>, signature (13,-70,202,-336,320,-160,32).
%F G.f.: (1-12*x+59*x^2-152*x^3+218*x^4-168*x^5+58*x^6-6*x^7) / ((1-x) *(1-2*x)^4 *(1-4*x+2*x^2)).
%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t CoefficientList[Series[(1 -12*x +59*x^2 -152*x^3 +218*x^4 -168*x^5 + 58*x^6 -6*x^7)/((1-x)*(1-2*x)^4*(1-4*x+2*x^2)), {x,0,50}], x] (* _G. C. Greubel_, Oct 22 2018 *)
%o (PARI)
%o gf=(1-12*x+59*x^2-152*x^3+218*x^4-168*x^5+58*x^6-6*x^7)/( (1-x)*(1-2*x)^4*(1-4*x+2*x^2) )
%o v165528=Vec(gf+O('x^66))
%o /* _Joerg Arndt_, Aug 16 2012 */
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 -12*x +59*x^2 -152*x^3 +218*x^4 -168*x^5 + 58*x^6 -6*x^7)/((1-x)*(1- 2*x)^4*(1-4*x+2*x^2)))); // _G. C. Greubel_, Oct 22 2018
%K nonn
%O 0,3
%A _Vincent Vatter_, Sep 21 2009
%E Prepended a(0)=1 by _Joerg Arndt_, Aug 16 2012
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