This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A257561 #32 Jul 05 2024 07:53:26
%S A257561 1,1,2,6,21,80,322,1346,5783,25372,113174,511649,2338988,10793251,
%T A257561 50205607,235156609,1108120540,5249646137,24987770893,119443412277,
%U A257561 573125649031,2759515312908,13328311926552,64559295743113,313530998739472,1526333617345412,7447070497787110,36409703715788374,178353171835771153,875224495042876048,4302111437028045585
%N A257561 Number of permutations of length n that avoid the patterns 4231, 4312, and 4321.
%C A257561 a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 2>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 second element is larger than the fourth element. - _Sergey Kitaev_, Dec 09 2020
%H A257561 Jay Pantone, <a href="/A257561/b257561.txt">Table of n, a(n) for n = 0..500</a>
%H A257561 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 A257561 D. Callan, T. Mansour, <a href="http://arxiv.org/abs/1705.00933">Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns</a>, arXiv:1705.00933 [math.CO] (2017), Table 1 No. 241.
%H A257561 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 A257561 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.
%F A257561 G.f. satisfies (2*x^2+8*x-1)*F(x)^4 + (x^3+4*x^2-46*x+5)*F(x)^3 + (3*x^3-21*x^2+94*x-9)*F(x)^2 + (x^3+12*x^2-82*x+7)*F(x) + 3*x^2+26*x-2 = 0. - _Jay Pantone_, Oct 01 2015
%F A257561 a(n) ~ (2*sqrt(phi) + phi^2)^n / (2*sqrt(Pi*c)*n^(3/2)), where phi = A001622 is the golden ratio and c = 0.8259440839165470204581761605617676911185302765... is the smallest positive real root of the equation 62742241 + 678297200*c - 490473522*c^2 - 749210300*c^3 + 314712204*c^4 - 33996440*c^5 + 1143417*c^6 - 1180*c^7 + c^8 = 0. - _Vaclav Kotesovec_, Jul 05 2024
%e A257561 a(4) = 21 because there are 24 permutations of length 4, and 3 of them do not avoid 4231, 4312, and 4321.
%K A257561 nonn
%O A257561 0,3
%A A257561 _Jay Pantone_, Apr 30 2015