A340568 Total number of consecutive triples matching the pattern 132 in all faro permutations of length n.

0, 0, 0, 1, 4, 10, 28, 61, 152, 318, 748, 1538, 3496, 7124, 15832, 32093, 70192, 141814, 306508, 617878, 1323272, 2663340, 5662600, 11383986, 24061264, 48330540, 101653368, 204049636, 427414672, 857503784, 1789891888, 3589478621, 7469802592, 14974962854, 31081371148

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. Also the popularity of consecutive pattern 213.

Examples

			For n = 4, there are 6 faro permutations: 1234, 1243, 1324, 2134, 2143, 3142. They contain in total 4 consecutive patterns 132 and also 4 consecutive patterns 213.
		

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, A340567, A340569.

Programs

PARI

seq(n)={my(t=sqrt(1-4*x^2+O(x^n))); Vec(x*(-1+4*x^2+2*x+(1-2*x)*t) / ((1-2*x)*(1+t)*t), -(1+n))} \\ Andrew Howroyd, Jan 11 2021

Formula

G.f.: x*(-1+4*x^2+2*x+(1-2*x)*sqrt(1-4*x^2)) / ((1-2*x)*(1+sqrt(1-4*x^2))*sqrt(1-4*x^2)).