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A340568
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Total number of consecutive triples matching the pattern 132 in all faro permutations of length n.
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4
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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
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OFFSET
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0,5
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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. Also the popularity of consecutive pattern 213.
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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.
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PROG
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(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
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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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