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A383406 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)}).

%I A383406 #5 Apr 26 2025 08:28:34
%S A383406 1,1,0,0,2,14,88,632,5152,46976,474056,5249064,63298724,825977620,
%T A383406 11597642568,174371083288,2795208188972,47592162832412,
%U A383406 857760977798888,16315057829100968,326599827759568812,6863964030561807340,151109048051281532488,3477542225297684400056,83503678542689445133052
%N A383406 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)}).
%C A383406 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 (0,0), (0,1), (1,0), (1,2), (2,1), and (2,2) shaded.
%H A383406 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 4.4 at page 17.
%F A383406 G.f.: (1 + t)^2 *A(t)/((1 + t)^2 + t^2*(A(t) - t - 1)*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 A383406 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383406 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 A383406 Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312.
%K A383406 nonn,easy
%O A383406 0,5
%A A383406 _Dan Li_, Apr 25 2025