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A340567
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Total number of ascents in all faro permutations of length n.
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4
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0, 0, 1, 4, 11, 26, 62, 134, 303, 634, 1394, 2872, 6206, 12676, 27068, 54994, 116423, 235706, 495722, 1001168, 2094714, 4223020, 8798756, 17715084, 36782246, 73980516, 153161332, 307808464, 635675228, 1276699336, 2630957432, 5281304554, 10863149303, 21797013946
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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)).
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EXAMPLE
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For n = 3 there are 3 faro permutations, namely 123, 213, 132. They contain 4 ascents (12, 23, 13 and 13) in total.
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PROG
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(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
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CROSSREFS
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A001405 counts faro permutations of length n.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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