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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 (list; graph; refs; listen; history; text; internal format)
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

Sequence in context: A193188 A216810 A065851 * A242790 A013044 A110577

Adjacent sequences:  A173935 A173936 A173937 * A173939 A173940 A173941

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

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Last modified October 16 10:55 EDT 2019. Contains 328056 sequences. (Running on oeis4.)