A228771 The number of skew sum indecomposable permutations which avoid the patterns 3124 and 4312.
1, 1, 3, 12, 53, 234, 1013, 4306, 18051, 74903, 308487, 1263393, 5152139, 20941298, 84897207, 343467388, 1387244237, 5595368133, 22543241377, 90739796783, 364954106877, 1466865660103, 5892463315373, 23658818086719, 94952826295865, 380947979933041, 1527871081396065, 6126157580638517, 24557525359295337, 98421154766829972
Offset: 1
Keywords
Examples
Example: a(4)=12 because there are 12 skew sum indecomposable permutations of length 4 which avoid the patterns 3124 and 4312.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], (2013)
Programs
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Mathematica
CoefficientList[Series[(1/x) (8 x^6 - 28 x^5 + 50 x^4 - 35 x^3 + 10 x^2 - Sqrt[-4 x + 1] (6 x^5 - 18 x^4 + 21 x^3 - 8 x^2 + x) - x) / (8 x^5 - 46 x^4 + 71 x^3 - 43 x^2 - Sqrt[-4 x + 1] (12 x^4 - 31 x^3 + 27 x^2 - 9 x + 1) + 11 x - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 09 2013 *)
Formula
G.f.: (8*x^6 - 28*x^5 + 50*x^4 - 35*x^3 + 10*x^2 - sqrt(-4*x + 1)*(6*x^5 - 18*x^4 + 21*x^3 - 8*x^2 + x) - x)/(8*x^5 - 46*x^4 + 71*x^3 - 43*x^2 - sqrt(-4*x + 1)*(12*x^4 - 31*x^3 + 27*x^2 - 9*x + 1) + 11*x - 1).
a(n) ~ 4^(n-1)/3 * (1+1/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: -163*(n+2)*(4*n-413) *a(n) +(-652*n^2-725425*n-452889) *a(n-1) +5*(14473*n^2+512276*n-443094) *a(n-2) +(-410045*n^2-2408964*n+8429009) *a(n-3) +2*(404156*n^2-1297075*n-1518084)*a(n-4) -8*(29333*n-32490)*(2*n-11)*a(n-5)=0. - R. J. Mathar, Jun 14 2016
Extensions
Corrected a(17) by Vincenzo Librandi, Sep 09 2013