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A277222
Number of permutations of length n that avoid 4123, 1324, and 31524.
2
1, 1, 2, 6, 22, 86, 343, 1374, 5497, 21926, 87176, 345612, 1366960, 5396604, 21275618, 83793268, 329791425, 1297428774, 5103039279, 20069815052, 78936685887, 310509337760, 1221686293926, 4807894697586, 18926706307263, 74529761736076, 293579002631157, 1156815016422286
OFFSET
0,3
LINKS
Sam Miner, Enumeration of several two-by-four classes, arXiv:1610.01908 [math.CO], 2016.
FORMULA
G.f.: (3 - 22*x + 54*x^2 - 54*x^3 + 25*x^4 - 4*x^5 - (1 - 6*x + 14*x^2 - 16*x^3 + 5*x^4)*sqrt(1 - 4*x))/(2 * sqrt(1 - 4*x) * (1 - 3*x + x^2)^2).
a(n) ~ 2^(2*n+4)/(25*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2016
EXAMPLE
For n = 4, a(n)=22, since all permutations of length 4 except 4123 and 1324 are included.
MATHEMATICA
CoefficientList[Series[(3 -22*x +54*x^2 -54*x^3 +25*x^4 -4*x^5 -(1 -6*x + 14*x^2 -16*x^3 +5*x^4)*Sqrt[1 -4*x])/(2*Sqrt[1 -4*x]*(1 -3*x + x^2)^2), {x, 0, 40}], x] (* G. C. Greubel, Apr 09 2017 *)
PROG
(PARI) x='x+O('x^40); Vec((3-22*x+54*x^2-54*x^3+25*x^4-4*x^5-(1-6*x + 14*x^2 -16*x^3 +5*x^4)*sqrt(1-4*x))/(2*sqrt(1-4*x)*(1-3*x+x^2)^2)) \\ G. C. Greubel, Apr 09 2017
(Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3 -22*x +54*x^2 -54*x^3 +25*x^4 -4*x^5 -(1 -6*x + 14*x^2 -16*x^3 +5*x^4)*Sqrt(1 -4*x))/(2*Sqrt(1 -4*x)*(1 -3*x + x^2)^2))); // G. C. Greubel, Oct 22 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Sam Miner, Oct 05 2016
STATUS
approved