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%I #54 Mar 28 2021 22:24:59
%S 1,1,2,6,22,90,396,1837,8864,44074,224352,1163724,6129840,32703074,
%T 176351644,959658200,5262988330,29057961666,161374413196,900792925199,
%U 5050924332096,28434661250454,160644331001476,910455895039056,5174722258676440,29486753617569684
%N Number of permutations of length n which avoid the patterns 1234 and 1324.
%C These permutations have an "enumeration scheme" of depth 4, see D. Zeilberger's article in the links.
%C G.f. conjectured to be non-D-finite (see Albert et al. link). - _Jay Pantone_, Oct 01 2015
%C a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 2>4, 3>4} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is the smallest. - _Sergey Kitaev_, Dec 10 2020
%H Andrew Baxter and Jay Pantone, <a href="/A053617/b053617.txt">Table of n, a(n) for n = 0..600</a> (terms n=1..100 from Andrew Baxter)
%H Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, Vincent Vatter, <a href="http://arxiv.org/abs/1510.00269">Generating Permutations with Restricted Containers</a>, arXiv:1510.00269 [math.CO], 2015.
%H Alice L. L. Gao, 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, 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 Kremer, Darla and Shiu, Wai Chee, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00042-6">Finite transition matrices for permutations avoiding pairs of length four patterns</a>, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Enumerations_of_specific_permutation_classes#Classes_avoiding_two_patterns_of_length_4">Permutation classes avoiding two patterns of length 4</a>.
%H D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/WILF.html">Enumeration schemes and more importantly their automatic generation</a>, Annals of Combinatorics 2 (1998) 185-195. The link is to an overview on Doron Zeilberger's home page; there is a local copy <a href="/A053617/a053617.pdf">here</a> [Pdf file only, no active links]
%Y Cf. A032351, A053614, A106228, A165542, A165545, A257561, A257562.
%K nonn
%O 0,3
%A _Moa Apagodu_, Mar 20 2000
%E More terms from _Andrew Baxter_, May 20 2011
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