OFFSET
0,4
COMMENTS
Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one.
LINKS
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Table 1.
FORMULA
G.f.: 2*x*(4*x^2 + x + sqrt(1 - 4*x^2) - 1)/((1 - 2*x)*sqrt(1 - 4*x^2)*(sqrt(1 - 4*x^2) + 1)).
EXAMPLE
For n = 3 there are 3 faro permutations, namely 123, 213, 132. They contain 4 ascents (12, 23, 13 and 13) in total.
PROG
(PARI) seq(n)={my(t=sqrt(1-4*x^2+O(x^n))); Vec(2*x*(4*x^2 + x + t - 1)/((1 - 2*x)*t*(t + 1)), -(1+n))} \\ Andrew Howroyd, Jan 11 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Sergey Kirgizov, Jan 11 2021
STATUS
approved