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 A383312 #12 May 09 2025 02:36:28
%S A383312 1,1,0,0,2,14,86,624,5096,46554,470446,5214936,62943852,821949042,
%T A383312 11548027442,173711893048,2785807179384,47448884653218,
%U A383312 855436571437710,16275060021803232,325872090863707740,6850004083354211050,150827444158572339810,3471582648001267649808,83371646323922972242776
%N A383312 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,1),(2,0),(2,2)}).
%C A383312 A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0<i<n. The sequence counts the number a(n) of king permutations of length n that avoid the mesh pattern 12 with squares (0,1), (0,2), (1,0), (1,1), (2,0), and (2,2) shaded.
%H A383312 Dan Li and Philip B. Zhang, <a href="https://arxiv.org/abs/2411.18131">Distributions of mesh patterns of short lengths on king permutations</a>, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.3 at page 16.
%F A383312 G.f.: (t + 1/(1 + t) - t^2*A(t)^2/((1 + t)*(1 + t + t*A(t))))*A(t) where A(t) = Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^n is the g.f. for king permutations given by A002464.
%e A383312 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383312 For n = 5 the a(5) = 14 solutions are these 14 permutations: 13524, 14253, 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
%Y A383312 Cf. A002464, A382644, A382645, A382651, A383040, A383107.
%K A383312 nonn,easy
%O A383312 0,5
%A A383312 _Dan Li_, Apr 22 2025