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Number of permutations of length n which avoid the patterns 4213 and 1432.
1

%I #26 Sep 08 2022 08:45:48

%S 1,1,2,6,22,87,352,1434,5861,24019,98677,406291,1676009,6924618,

%T 28646875,118638038,491765865,2039944740,8467475533,35166107745,

%U 146115418937,607353499821,2525443862594,10504254304765,43702642447260,181865873468907,756979080521743,3151341504417932

%N Number of permutations of length n which avoid the patterns 4213 and 1432.

%H G. C. Greubel, <a href="/A165533/b165533.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.: 1 + x*(1 - x)*(1 - 2*x)*(1 - 7*x + 17*x^2 - 16*x^3 + 4*x^4 + (1 - 3*x + 3*x^2)*sqrt(1 - 4*x))/(2 - 22*x + 96*x^2 - 220*x^3 + 282*x^4 - 196*x^5 + 64*x^6 - 8*x^7). - _G. C. Greubel_, Oct 22 2018

%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.

%t CoefficientList[Series[1 + x*(1 - x)*(1 - 2*x)*(1 - 7*x + 17*x^2 - 16*x^3 + 4*x^4 + (1 - 3*x + 3*x^2)*Sqrt[1 - 4*x])/(2 - 22*x + 96*x^2 - 220*x^3 + 282*x^4 - 196*x^5 + 64*x^6 - 8*x^7), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 22 2018 *)

%o (PARI) x='x+O('x^50); Vec(1 + x*(1-x)*(1-2*x)*(1-7*x+17*x^2-16*x^3+4*x^4 + (1-3*x+3*x^2)*sqrt(1-4*x))/(2-22*x+96*x^2-220*x^3+282*x^4-196*x^5 + 64*x^6-8*x^7)) \\ _G. C. Greubel_, Oct 22 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1 + x*(1-x)*(1-2*x)*(1-7*x+17*x^2-16*x^3+4*x^4 +(1-3*x + 3*x^2)*Sqrt(1 - 4*x))/(2-22*x+96*x^2-220*x^3+282*x^4-196*x^5+64*x^6-8*x^7))); // _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