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

Author

Sergey Kirgizov, Jan 12 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 = 4, there are 6 faro permutations: 1234, 1243, 1324, 2134, 2143, 3142. They contain in total 4 consecutive patterns 123.
		

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, A340568.

Programs

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 *)

Formula

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