A383040 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(2,0)}).

1, 1, 0, 0, 2, 12, 78, 568, 4674, 42944, 436314, 4860020, 58914870, 772330276, 10888803374, 164310553184, 2642525580218, 45124440536632, 815438318526482, 15547317496485932, 311912067538692126, 6568399090178800988, 144867849880285518694, 3339550150164041194232, 80315480372245746015970

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) = 12 solutions are these 12 permutations: 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 4.1 at page 13.

Crossrefs

Cf. A002464, A382644, A382645, A382651.

Formula

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