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A383434 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,2),(1,0),(1,1),(2,0),(2,1)}).

%I A383434 #6 May 03 2025 21:42:32
%S A383434 1,1,0,0,2,10,68,500,4170,38730,397172,4459116,54421082,717571442,
%T A383434 10167743668,154104395348,2487968793386,42630767594522,
%U A383434 772730550801940,14773475294401180,297121458577213850,6270996358146824738,138591948457411817684,3200867594024256790020,77112844928711640695594
%N A383434 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,2),(1,0),(1,1),(2,0),(2,1)}).
%C A383434 A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0<i<n. a(n) is the number of king permutations of length n that avoid the mesh pattern 12 with squares (0,2), (1,0), (1,1), (2,0), and (2,1) shaded.
%H A383434 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 4.8 at page 23.
%F A383434 G.f.: 1 + t + 1/(1 + 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 A383434 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383434 For n = 5 the a(5) = 10 solutions are these 10 permutations: 24153, 25314, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
%Y A383434 Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406, A383407, A383408, A383433.
%K A383434 nonn,easy
%O A383434 0,5
%A A383434 _Dan Li_, Apr 27 2025