login
Number of Wachs permutations of size n.
2

%I #33 Dec 21 2023 20:55:22

%S 1,1,2,4,8,24,48,192,384,1920,3840,23040,46080,322560,645120,5160960,

%T 10321920,92897280,185794560,1857945600,3715891200,40874803200,

%U 81749606400,980995276800,1961990553600,25505877196800,51011754393600,714164561510400,1428329123020800

%N Number of Wachs permutations of size n.

%C A Wachs permutation pi is a permutation of [n] such that |pi^{-1}(i) - pi^{-1}(i*)| <= 1, for all 1 <= i <= n-1, where i* is defined as i-1 if i is even, i+1 if i is odd and i+1 <= n, and n otherwise.

%H Alois P. Heinz, <a href="/A359039/b359039.txt">Table of n, a(n) for n = 0..806</a>

%H Francesco Brenti and Paolo Sentinelli, <a href="https://arxiv.org/abs/2212.04932">Wachs permutations, Bruhat order and weak order</a>, arXiv:2212.04932 [math.CO], 2022.

%F If n=2m, then a(n) = m!*2^m, if n=2m+1, then a(n) = (m+1)!*2^m.

%F a(n) = A081123(n+1)*A016116(n). - _Alois P. Heinz_, Jan 23 2023

%F Sum_{n>=0} 1/a(n) = 3*sqrt(e) - 2. - _Amiram Eldar_, Jan 25 2023

%F D-finite with recurrence a(n) +2*a(n-1) +(-n-1)*a(n-2) +2*(-n+1)*a(n-3)=0. - _R. J. Mathar_, Jul 17 2023

%e For n=4, a(n)=8, since we have the 8 Wachs permutations 1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.

%p A359039 := proc(n)

%p local m ;

%p m := floor(n/2) ;

%p if type(n,'even') then

%p m!*2^m ;

%p else

%p (m+1)!*2^m ;

%p end if;

%p end proc: # _R. J. Mathar_, Jul 17 2023

%p # second Maple program:

%p a:= n-> ceil(n/2)!*2^floor(n/2):

%p seq(a(n), n=0..28); # _Alois P. Heinz_, Dec 21 2023

%t a[n_]:=If[EvenQ[n], (n/2)! 2^(n/2), ((n + 1)/2)!*2^((n - 1)/2)]

%Y Cf. A016116, A081123.

%K nonn

%O 0,3

%A _Per W. Alexandersson_, Dec 13 2022