cp's OEIS Frontend

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

%I A382651 #20 Apr 09 2025 21:14:12
%S A382651 1,0,0,0,2,10,68,500,4174,38770,397544,4462476,54452394,717877882,
%T A382651 10170925492,154139627692,2488385952526,42636054584106,
%U A382651 772802263942376,14774515232543556,297137552306148570,6271261537872652418,138596588342412866276,3200953561821628327956,77114526810424117688014
%N A382651 Number of king permutations on n elements without strict fixed points.
%C A382651 A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0<i<n. a(n) counts king permutations of length n that have no strict fixed points (that is, that avoid the mesh pattern of length 1 with squares (0,1) and (1,0) shaded).
%H A382651 Dan Li and Philip B. Zhang, <a href="https://arxiv.org/abs/2411.18131">Distributions of mesh patterns of short lengths on king permutations</a>, arXiv:2411.18131 [math.CO], 2024. See formula (7) at page 6.
%F A382651 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.
%F A382651 a(n) ~ exp(-2) * n!. - _Vaclav Kotesovec_, Apr 04 2025
%e A382651 a(4) = 2 corresponds to these two permutations: 2413, 3142.
%e A382651 a(5) = 10 corresponds to these 10 permutations of length 5: 24153, 25314, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
%t A382651 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 *)
%o A382651 (PARI)
%o A382651 N=30; t='t+O('t^N);
%o A382651 A=sum(n=0, N, n!*t^n*(1-t)^n/(1+t)^n);
%o A382651 gf=(1 + t)*A/(1 + t + t*A);
%o A382651 Vec(gf)  \\ _Joerg Arndt_, Apr 03 2025
%Y A382651 Cf. A002464, A382644, A382645.
%K A382651 nonn
%O A382651 0,5
%A A382651 _Dan Li_, Apr 02 2025