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

Author

Jay Pantone, Sep 08 2013

Keywords

• nonn

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)

Crossrefs

A228771(n) = A165534(n) - A228769(n)

Programs

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