A340567 Total number of ascents in all faro permutations of length n.

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

Offset: 0

Author

Sergey Kirgizov, Jan 11 2021

Keywords

• nonn

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.

Examples

			For n = 3 there are 3 faro permutations, namely 123, 213, 132. They contain 4 ascents (12, 23, 13 and 13) in total.
		

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.

Crossrefs

A001405 counts faro permutations of length n.

Cf. A107373 (descents), A340568, A340569.

Programs

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

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)).