A165538 Number of permutations of length n which avoid the patterns 4312 and 3142.
1, 1, 2, 6, 22, 88, 367, 1568, 6810, 29943, 132958, 595227, 2683373, 12170778, 55499358, 254297805, 1170248190, 5406570910, 25068420955, 116617923611, 544157590706, 2546278167018, 11945937322413, 56180864428301, 264812677643417, 1250853429148333, 5920145717412047
Offset: 0
Keywords
Examples
There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
Links
- M. H. Albert, M. D. Atkinson, and V. Vatter, Inflations of geometric grid classes: three case studies, arXiv:1209.0425 [math.CO], 2012.
- Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
- Christian Bean, Émile Nadeau, Henning Ulfarsson, Enumeration of Permutation Classes and Weighted Labelled Independent Sets, arXiv:1912.07503 [math.CO], 2019.
- Robert Brignall, Jakub Sliacan, Juxtaposing Catalan permutation classes with monotone ones, arXiv:1611.05370 [math.CO], 2016.
- Juan B. Gil, Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018.
- 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.
- Wikipedia, Permutation classes avoiding two patterns of length 4.
Programs
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Mathematica
CoefficientList[Series[(1 + Sqrt[1 - 4*x]) / (4*x) - Sqrt[2*(1 + Sqrt[1 - 4*x] - 2*x)*(1 - x)*(1 - 5*x)] / (4*(1-x)*x), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 06 2024 *)
Formula
G.f. f satisfies: (x^3-2*x^2+x)*f^4+(4*x^3-9*x^2+6*x-1)*f^3+(6*x^3-12*x^2+7*x-1)*f^2+(4*x^3-5*x^2+x)*f+x^3 = 0.
From Vaclav Kotesovec, Jul 06 2024: (Start)
G.f.: (1 + sqrt(1-4*x)) / (4*x) - sqrt(2*(1 + sqrt(1-4*x)-2*x)*(1-x)*(1-5*x)) / (4*(1-x)*x).
a(n) ~ (1 + sqrt(5)) * 5^(n+1) / (16 * sqrt(Pi) * n^(3/2)). (End)
Extensions
Reference corrected by Vincent Vatter, Sep 04 2012
a(0)=1 prepended by Alois P. Heinz, Jul 06 2024