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A383408 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,2),(1,0),(1,1),(1,2),(2,1)}).

%I A383408 #5 Apr 26 2025 08:28:38
%S A383408 1,1,0,0,2,14,88,632,5152,46972,474008,5248616,63294680,825940168,
%T A383408 11597278752,174367336624,2795167052832,47591679875632,
%U A383408 857754907053056,16314976128578752,326598651690933216,6863945954213702816,151108752072042907968,3477537076217415673344,83503583639127861347392
%N A383408 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,2),(1,0),(1,1),(1,2),(2,1)}).
%C A383408 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), (0,2), (1,0), (1,1), (1,2), and (2,1) shaded.
%H A383408 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 4.6 at page 19.
%F A383408 G.f.: (1 + t)*(A(t) - t)/(1 + t*(A(t) - t - 1)) 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 A383408 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383408 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 A383408 Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406, A383407.
%K A383408 nonn,easy
%O A383408 0,5
%A A383408 _Dan Li_, Apr 26 2025