OFFSET
0,3
COMMENTS
These permutations have an enumeration scheme of depth 5.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
Sam Miner, Enumeration of several two-by-four classes, arXiv preprint arXiv:1610.01908 [math.CO], 2016.
V. Vatter, Enumeration schemes for restricted permutations, Combin., Prob. and Comput. 17 (2008), 137-159
FORMULA
G.f.: (2 - 8*x + 2*x^2 + 17*x^3 - 15*x^4 + 4*x^5 - (2 - 10*x + 16*x^2 - 9*x^3 + 2*x^4)*sqrt(1 - 6*x + 5*x^2))/(2*x*(2 - x)^2*(1 - 2*x)*(1 - 3*x + x^2)). - G. C. Greubel, Oct 22 2018
EXAMPLE
There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
MATHEMATICA
CoefficientList[Series[(2 - 8*x + 2*x^2 + 17*x^3 - 15*x^4 + 4*x^5 - (2 - 10*x + 16*x^2 - 9*x^3 + 2*x^4)*Sqrt[1 - 6*x + 5*x^2])/(2*x*(2 - x)^2*(1 - 2*x)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* G. C. Greubel, Oct 22 2018 *)
PROG
(PARI) x='x+O('x^30); Vec((2 - 8*x + 2*x^2 + 17*x^3 - 15*x^4 + 4*x^5 - (2 - 10*x + 16*x^2 - 9*x^3 + 2*x^4)*sqrt(1 - 6*x + 5*x^2))/(2*x*(2 - x)^2*(1 - 2*x)*(1 - 3*x + x^2))) \\ G. C. Greubel, Oct 22 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((2 - 8*x + 2*x^2 + 17*x^3 - 15*x^4 + 4*x^5 - (2 - 10*x + 16*x^2 - 9*x^3 + 2*x^4)*Sqrt(1 - 6*x + 5*x^2))/(2*x*(2 - x)^2*(1 - 2*x)*(1 - 3*x + x^2)))); // G. C. Greubel, Oct 22 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vincent Vatter, Sep 21 2009
EXTENSIONS
Terms a(21) onward added by G. C. Greubel, Oct 22 2018
STATUS
approved