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A165535
Number of permutations of length n which avoid the patterns 4231 and 3124.
1
1, 1, 2, 6, 22, 88, 363, 1508, 6255, 25842, 106327, 435965, 1782733, 7275351, 29648647, 120707058, 491113791, 1997372920, 8121565606, 33020039047, 134248625367, 545835561195, 2219474787024, 9025797884775, 36709145207578, 149320519008554, 607466672855393
OFFSET
0,3
REFERENCES
Kremer, Darla and Shiu, Wai Chee; Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
FORMULA
G.f.: 1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x+2*x^2)*sqrt(1-4*x)) / (2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3)).
EXAMPLE
There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
MATHEMATICA
CoefficientList[Series[1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x+ 2*x^2 )*Sqrt[1-4*x])/(2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3)), {x, 0, 30}], x] (* G. C. Greubel, Oct 22 2018 *)
PROG
(PARI) x='x+O('x^30); Vec(1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x +2*x^2)*sqrt(1-4*x))/(2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3))) \\ G. C. Greubel, Oct 22 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x+2*x^2)*Sqrt(1-4*x)) / (2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3)))); // G. C. Greubel, Oct 22 2018
CROSSREFS
Sequence in context: A363811 A150263 A165534 * A319028 A165536 A032351
KEYWORD
nonn,easy
AUTHOR
Vincent Vatter, Sep 21 2009
EXTENSIONS
More terms, g.f., and reference by Vincent Vatter, Sep 04 2012
a(0)=1 prepended by Alois P. Heinz, Feb 18 2016
STATUS
approved