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)}).

1, 1, 0, 0, 2, 14, 88, 636, 5174, 47122, 475128, 5257976, 63381078, 826817350, 11607019144, 174484968604, 2796703640190, 47613279070594, 858080079253440, 16320196781972904, 326687694661023774, 6865553778933359142, 151139392725808178080, 3478151644016630307452, 83516524547918673461238

Offset: 0

Author

Dan Li, Apr 22 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 4.2 at page 15.

Crossrefs

Cf. A002464, A382644, A382645, A382651, A383040.

Formula

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.

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