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%I #17 Jan 11 2021 23:16:24
%S 0,0,1,4,11,26,62,134,303,634,1394,2872,6206,12676,27068,54994,116423,
%T 235706,495722,1001168,2094714,4223020,8798756,17715084,36782246,
%U 73980516,153161332,307808464,635675228,1276699336,2630957432,5281304554,10863149303,21797013946
%N Total number of ascents in all faro permutations of length n.
%C 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.
%H Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, <a href="https://arxiv.org/abs/2010.06270">Pattern statistics in faro words and permutations</a>, arXiv:2010.06270 [math.CO], 2020. See Table 1.
%F 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)).
%e For n = 3 there are 3 faro permutations, namely 123, 213, 132. They contain 4 ascents (12, 23, 13 and 13) in total.
%o (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
%Y A001405 counts faro permutations of length n.
%Y Cf. A107373 (descents), A340568, A340569.
%K nonn
%O 0,4
%A _Sergey Kirgizov_, Jan 11 2021
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