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A340567 Total number of ascents in all faro permutations of length n. 4
0, 0, 1, 4, 11, 26, 62, 134, 303, 634, 1394, 2872, 6206, 12676, 27068, 54994, 116423, 235706, 495722, 1001168, 2094714, 4223020, 8798756, 17715084, 36782246, 73980516, 153161332, 307808464, 635675228, 1276699336, 2630957432, 5281304554, 10863149303, 21797013946 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..33.

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

A001405 counts faro permutations of length n.

Cf. A107373 (descents), A340568, A340569.

Sequence in context: A034334 A036891 A183276 * A268775 A294840 A014630

Adjacent sequences:  A340564 A340565 A340566 * A340568 A340569 A340570

KEYWORD

nonn

AUTHOR

Sergey Kirgizov, Jan 11 2021

STATUS

approved

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Last modified July 26 16:44 EDT 2021. Contains 346294 sequences. (Running on oeis4.)