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A173938
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The number of permutations avoiding simultaneously consecutive patterns 123 and 231.
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2
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1, 1, 2, 4, 11, 39, 161, 784, 4368, 27260, 189540, 1448860, 12076408, 109102564, 1061259548, 11060323280, 122963473024, 1452414435968, 18164949751872, 239807221886128, 3332441297971360, 48624372236312912, 743273838888233264, 11878134680411900928
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OFFSET
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0,3
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COMMENTS
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Terms a(11) through a(14) calculated by Elizalde and Noy, who state that an involved explicit form for the e.g.f. can be found in terms of integrals containing the error function.
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REFERENCES
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S. Elizalde and M. Noy, Consecutive patterns in permutations (Theorem 5.1), Adv. Appl. Math. 30 (2003) 110-125.
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LINKS
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S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, Ars Combinatoria 76 (2005), 321-350 [MR2152770]
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FORMULA
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For all n >= 3, A(n) = a(n-1) + a(n;1) + a(n;2) + ... + a(n;n-1), where for all 1<= i <= n, a(n;i)= Sum_{j=1..i-1} a(n-1;j) + Sum_{j=i..n-2} (n-1-j)*a*(n-2;j), and a(3;1)=1, a(3;2)=1 a(3;3)=2.
a(n) ~ c * d^n * n!, where d = A246041 = 0.6948193008667305362671927506... is the root of the equation sqrt(2*Pi)*(erfi(1/sqrt(2)) + erfi((1/d-1)/sqrt(2))) = 2*exp(1/2), c = 1.991594102047693697258367189... . - Vaclav Kotesovec, Aug 23 2014
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EXAMPLE
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Example: For n = 3 a(3) = 4 since 132, 213, 312, and 321 are the 3-permutations avoiding 123 and 231.
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, 0), j=1..`if`(t>0, min(u, t-1), u))+
`if`(t>0, 0, add(b(u+j-1, o-j, j), j=1..o)))
end:
a:= n-> b(n, 0, 0):
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MATHEMATICA
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FullSimplify[CoefficientList[Series[1 + Integrate[(2*Sqrt[E]/(2*Sqrt[E] - Sqrt[2*Pi]*Erfi[1/Sqrt[2]] - Sqrt[2*Pi] * Erfi[(-1+x)/Sqrt[2]]))*((E^(1/2*(-1+x)^2) * (2 + Sqrt[2*E*Pi]*Erf[1/Sqrt[2]] - Pi*Erf[1/Sqrt[2]]*Erfi[1/Sqrt[2]] + Erf[(-1+x)/Sqrt[2]]*(Sqrt[2*E*Pi] - Pi*Erfi[1/Sqrt[2]]) + HypergeometricPFQ[{1, 1}, {3/2, 2}, -1/2] - (-1+x)^2 * HypergeometricPFQ[{1, 1}, {3/2, 2}, -1/2*(-1+x)^2])) / (2*Sqrt[E] - Sqrt[2*Pi]*(Erfi[1/Sqrt[2]] + Erfi[(-1+x)/Sqrt[2]]))), x], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 22 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
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EXTENSIONS
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STATUS
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approved
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