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

1,3

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)

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

Example

Example: a(4)=12 because there are 12 skew sum indecomposable permutations of length 4 which avoid the patterns 3124 and 4312.

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 *)

Crossrefs

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

Sequence in context: A299113 A124202 A138269 * A370023 A151198 A151199

Adjacent sequences: A228768 A228769 A228770 * A228772 A228773 A228774

Keyword

nonn

Author

Jay Pantone, Sep 08 2013

Extensions

Corrected a(17) by Vincenzo Librandi, Sep 09 2013

Status

approved