A382651 Number of king permutations on n elements without strict fixed points.

1, 0, 0, 0, 2, 10, 68, 500, 4174, 38770, 397544, 4462476, 54452394, 717877882, 10170925492, 154139627692, 2488385952526, 42636054584106, 772802263942376, 14774515232543556, 297137552306148570, 6271261537872652418, 138596588342412866276, 3200953561821628327956, 77114526810424117688014

Offset: 0

Author

Dan Li, Apr 02 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

			a(4) = 2 corresponds to these two permutations: 2413, 3142.
a(5) = 10 corresponds to these 10 permutations of length 5: 24153, 25314, 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 formula (7) at page 6.

Crossrefs

Cf. A002464, A382644, A382645.

Programs

Mathematica

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

PARI

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

Formula

G.f.: (1 + t)*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.

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

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