A383406 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)}).

1, 1, 0, 0, 2, 14, 88, 632, 5152, 46976, 474056, 5249064, 63298724, 825977620, 11597642568, 174371083288, 2795208188972, 47592162832412, 857760977798888, 16315057829100968, 326599827759568812, 6863964030561807340, 151109048051281532488, 3477542225297684400056, 83503678542689445133052

Offset: 0

Author

Dan Li, Apr 25 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 4.4 at page 17.

Crossrefs

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

Formula

G.f.: (1 + t)^2 *A(t)/((1 + t)^2 + t^2*(A(t) - 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.

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