A340569 Total number of consecutive triples matching the pattern 123 in all faro permutations of length n.

0, 0, 0, 1, 4, 10, 24, 53, 116, 246, 520, 1082, 2248, 4628, 9520, 19469, 39796, 81022, 164904, 334670, 679064, 1374924, 2783440, 5625666, 11368904, 22945820, 46307664, 93358228, 188202256, 379078952, 763506784, 1536708413, 3092806516, 6220970702, 12512656744, 25154958278

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.

Links

Table of n, a(n) for n=0..35.

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+2*x) * (1-sqrt(1-4*x^2)) / ((1-2*x) * (1+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 123.

Mathematica

Table[SeriesCoefficient[x*(1+2*x)*(1-Sqrt[1-4*x^2])/((1-2*x) * (1+Sqrt[1-4*x^2])), {x, 0, n}], {n, 0, 35}] (* Stefano Spezia, Jan 12 2021 *)

Crossrefs

A001405 counts faro permutations of length n.

Cf. A107373, A340567, A340568.

Sequence in context: A280541 A080615 A173729 * A097976 A279851 A266367

Adjacent sequences: A340566 A340567 A340568 * A340570 A340571 A340572

Keyword

nonn

Author

Sergey Kirgizov, Jan 12 2021

Status

approved