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%I #39 Jul 06 2024 12:03:46
%S 1,1,2,6,22,88,367,1571,6861,30468,137229,625573,2881230,13388094,
%T 62688448,295504025,1401195334,6678877732,31984089193,153809536017,
%U 742462191363,3596290278723,17473993136316,85147347832182,415997039428899,2037323575386383
%N Number of permutations of length n which avoid the patterns 4213 and 3421.
%H Alois P. Heinz, <a href="/A165539/b165539.txt">Table of n, a(n) for n = 0..1000</a>
%H David Bevan, <a href="http://arxiv.org/abs/1407.0570">The permutation classes Av(1234, 2341) and Av(1243, 2314)</a>, arXiv:1407.0570 [math.CO], 2014.
%H Kremer, Darla and Shiu, Wai Chee; <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>.
%F G.f.: F = F(z) has minimal polynomial (z-3*z^2+2*z^3) - (1-5*z+8*z^2-5*z^3)*F + (2*z-5*z^2+4*z^3)*F^2 + z^3*F^3. - _David Bevan_, Jun 23 2014
%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t terms = 25;
%t F[_] = 0; Do[F[z_] = (z(1 - 3z + 2z^2 + z^2 F[z]^3 + (2 - 5z + 4z^2) F[z]^2 )) / (1 - 5z + 8z^2 - 5z^3) + O[z]^(terms+1), {terms+1}];
%t CoefficientList[F[z], z] // Rest (* _Jean-François Alcover_, Nov 25 2018 *)
%K nonn,changed
%O 0,3
%A _Vincent Vatter_, Sep 21 2009
%E More terms from _David Bevan_, Feb 11 2014
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