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A173938
The number of permutations avoiding simultaneously consecutive patterns 123 and 231.
2
1, 1, 2, 4, 11, 39, 161, 784, 4368, 27260, 189540, 1448860, 12076408, 109102564, 1061259548, 11060323280, 122963473024, 1452414435968, 18164949751872, 239807221886128, 3332441297971360, 48624372236312912, 743273838888233264, 11878134680411900928
OFFSET
0,3
COMMENTS
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.
REFERENCES
S. Elizalde and M. Noy, Consecutive patterns in permutations (Theorem 5.1), Adv. Appl. Math. 30 (2003) 110-125.
LINKS
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, Ars Combinatoria 76 (2005), 321-350 [MR2152770]
FORMULA
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
EXAMPLE
Example: For n = 3 a(3) = 4 since 132, 213, 312, and 321 are the 3-permutations avoiding 123 and 231.
MAPLE
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):
seq(a(n), n=0..25); # Alois P. Heinz, Oct 25 2013
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A193188 A216810 A065851 * A242790 A013044 A110577
KEYWORD
nonn
AUTHOR
Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
EXTENSIONS
a(15)-a(23) from Alois P. Heinz, Oct 25 2013
STATUS
approved