OFFSET
1,2
COMMENTS
These permutations have an enumeration scheme of depth 6.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Bevan, The permutation classes Av(1234,2341) and Av(1243,2314), arXiv:1407.0570 [math.CO], 2014.
Darla Kremer, and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
V. Vatter, Enumeration schemes for restricted permutations, Combin., Prob. and Comput. 17 (2008), 137-159.
FORMULA
G.f.: ((2-10*z+9*z^2+7*z^3-4*z^4)*sqrt(1-4*z) - (2-16*z+41*z^2-39*z^3+12*z^4)) / ((1-4*z)*(1-3*z+z^2)*((1-z)*sqrt(1-4*z) + (1-3*z))). - David Bevan, Jun 23 2014
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Aug 23 2014
Conjecture: +(14681*n+187954)*(n+3) *a(n) +(14681*n^2-2696783*n-4897218) *a(n-1) +(-888761*n^2+12771539*n+5490342) *a(n-2) +2*(1635223*n^2-14850835*n+13281012) *a(n-3) +2*(-1908503*n^2+15402653*n-25889820) *a(n-4) +156*(2*n-7)*(3301*n-14374) *a(n-5)=0. - R. J. Mathar, Jun 14 2016
EXAMPLE
There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
MATHEMATICA
Rest[CoefficientList[ Series[ ((2-10z+9z^2+7z^3-4z^4) Sqrt[1-4z] -(2-16z+41z^2-39z^3+12z^4)) / ((1-4z) (1-3z+z^2) ((1-z) Sqrt[1-4z] +(1-3z))), {z, 0, 40}], z]] (* David Bevan, Jun 23 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vincent Vatter, Sep 21 2009
EXTENSIONS
More terms from David Bevan, Jun 23 2014
STATUS
approved