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
Alois P. Heinz, Table of n, a(n) for n = 0..200
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, Ars Combinatoria 76 (2005), 321-350 [MR2152770]
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns
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
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