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

0,5

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.

Links

Table of n, a(n) for n=0..34.

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.: 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)).

Example

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.

Prog

(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

Crossrefs

A001405 counts faro permutations of length n.

Cf. A107373, A340567, A340569.

Sequence in context: A174938 A032248 A092504 * A038579 A133726 A026534

Adjacent sequences: A340565 A340566 A340567 * A340569 A340570 A340571

Keyword

nonn

Author

Sergey Kirgizov, Jan 11 2021

Status

approved