A383312 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,1),(2,0),(2,2)}).

1, 1, 0, 0, 2, 14, 86, 624, 5096, 46554, 470446, 5214936, 62943852, 821949042, 11548027442, 173711893048, 2785807179384, 47448884653218, 855436571437710, 16275060021803232, 325872090863707740, 6850004083354211050, 150827444158572339810, 3471582648001267649808, 83371646323922972242776

Offset

0,5

Comments

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 (0,1), (0,2), (1,0), (1,1), (2,0), and (2,2) shaded.

Links

Table of n, a(n) for n=0..24.

Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.3 at page 16.

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A037563 A005610 A065355 * A162478 A348615 A380552

Adjacent sequences: A383309 A383310 A383311 * A383313 A383314 A383315

Keyword

nonn,easy

Author

Dan Li, Apr 22 2025

Status

approved