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)}).

1, 1, 0, 0, 2, 14, 88, 636, 5174, 47122, 475124, 5257936, 63380706, 826813990, 11606987816, 174484661916, 2796700455414, 47613243806514, 858079661762692, 16320191491499712, 326687622910353650, 6865552738575268502, 151139376627154723752, 3478151378775992816412, 83516519907235226131286

Offset: 0

Author

Dan Li, Apr 26 2025

Keywords

• nonn
• easy

Comments

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

Examples

			For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
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.
		

Links

• Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.5 at page 18.

Crossrefs

Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406.

Formula

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.

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