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A383107 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).

%I A383107 #21 Apr 25 2025 15:13:08
%S A383107 1,1,0,0,2,14,88,636,5174,47122,475128,5257976,63381078,826817350,
%T A383107 11607019144,174484968604,2796703640190,47613279070594,
%U A383107 858080079253440,16320196781972904,326687694661023774,6865553778933359142,151139392725808178080,3478151644016630307452,83516524547918673461238
%N A383107 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).
%C A383107 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), (0,2), (1,0), (1,2), (2,0), and (2,1) shaded.
%H A383107 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 4.2 at page 15.
%F A383107 G.f.: (1/(1 + t) + t*(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 A383107 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383107 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 A383107 Cf. A002464, A382644, A382645, A382651, A383040.
%K A383107 nonn,easy
%O A383107 0,5
%A A383107 _Dan Li_, Apr 22 2025