A382645 Number of king permutations on n elements not beginning with the smallest element and not ending with the largest element.

1, 0, 0, 0, 2, 10, 68, 500, 4174, 38774, 397584, 4462848, 54455754, 717909202, 10171232060, 154142811052, 2488421201446, 42636471916622, 772807552752712, 14774586965277816, 297138592463202402, 6271277634164008170, 138596853553771517492, 3200958202120445923684, 77114612783976599209598

Offset: 0

Author

Dan Li, Apr 01 2025

Keywords

• nonn

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) = 10 solutions are these 10 permutations: 24153, 25314, 31524, 35142, 35241, 41352, 42513, 42531, 52413, and 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 formula (3) at page 5.

Crossrefs

Cf. A002464, A382644.

Programs

Mathematica

nmax = 20; CoefficientList[Series[x/(1+x) + Sum[k!*x^k*(1-x)^k/(1+x)^(k+2), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 04 2025 *)

PARI

my(N=30, t='t+O('t^N)); Vec(t/(1+t)+sum(n=0,N,n!*t^n*(1-t)^n/(1+t)^(n+2))) \\ Joerg Arndt, Apr 03 2025

Formula

G.f.: t/(1+t) + Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^(n+2).

a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Apr 04 2025

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