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Leading diagonal of triangle in A211321.
2

%I #45 Jul 24 2024 09:20:45

%S 1,1,2,6,24,116,632,3720,23072,148528,983072,6647776,45727616,

%T 318947136,2250473344,16034726016,115204555264,833701020416,

%U 6071393452544,44460988694016,327199429228544,2418586647786496,17948732704159744,133679253739800576

%N Leading diagonal of triangle in A211321.

%C Conjectured to be the number of permutations of length n avoiding the partially ordered pattern (POP) {1>5, 2>5, 5>3, 5>4} of length 5. That is, conjectured to be the number of length n permutations having no subsequences of length 5 in which the fifth element is larger than the elements in positions 3 and 4, but smaller than the elements in positions 1 and 2. - _Sergey Kitaev_, Dec 13 2020

%C a(n) is also the number of inversion sequences of length n that avoid the patterns 201 and 210. - _Jay Pantone_, Oct 11 2023

%C The conjecture of Kitaev has been proven. It can be restated as the number of size n permutations avoiding 45123, 45213, 54123, 54213. There are nineteen sets of permutations avoiding four size five permutations that are known to match this sequence. A further four are conjectured to match this sequence. - _Christian Bean_, Jul 23 2024

%H Jay Pantone, <a href="/A212198/b212198.txt">Table of n, a(n) for n = 0..200</a>

%H Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, <a href="https://permpal.com/perms/search_params/?we_type=cfs&amp;wilf_equivalence=45123%2C+45213%2C+54123%2C+54213">PermPAL database</a>.

%H Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, <a href="https://doi.org/10.37236/12686">Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding</a>, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); <a href="https://arxiv.org/abs/2312.07716">arXiv preprint</a>, arXiv:2312.07716 [math.CO], 2023.

%H Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.

%H Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016 [Section 2.29].

%H Chunyan Yan and Zhicong Lin, <a href="https://arxiv.org/abs/1912.03674">Inversion sequences avoiding pairs of patterns</a>, arXiv:1912.03674 [math.CO], 2019.

%F G.f.: (2-x-x*(1-8*x)^(1/2))/(4*x^2-4*x+2). - _Jay Pantone_, Oct 11 2023

%F a(n) ~ 2^(3*n) / (25*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 22 2024

%t CoefficientList[Series[(2 - x - x*(1 - 8*x)^(1/2))/(4*x^2 - 4*x + 2), {x, 0, 23}], x] (* _Michael De Vlieger_, Dec 19 2023 *)

%Y Cf. A211321.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, May 04 2012

%E More terms from _Jay Pantone_, Oct 11 2023