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A383407 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).

%I A383407 #5 Apr 26 2025 08:28:43
%S A383407 1,1,0,0,2,14,88,636,5174,47122,475124,5257936,63380706,826813990,
%T A383407 11606987816,174484661916,2796700455414,47613243806514,
%U A383407 858079661762692,16320191491499712,326687622910353650,6865552738575268502,151139376627154723752,3478151378775992816412,83516519907235226131286
%N A383407 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).
%C A383407 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), (1,2), (2,0), and (2,1) shaded.
%H A383407 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 4.5 at page 18.
%F A383407 G.f.: (1 + t)*(1 + t + t*(2 + t)*A(t))*A(t)/(1 + t + t*A(t))^2 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 A383407 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383407 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 A383407 Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406.
%K A383407 nonn,easy
%O A383407 0,5
%A A383407 _Dan Li_, Apr 26 2025