A359039 Number of Wachs permutations of size n.

1, 1, 2, 4, 8, 24, 48, 192, 384, 1920, 3840, 23040, 46080, 322560, 645120, 5160960, 10321920, 92897280, 185794560, 1857945600, 3715891200, 40874803200, 81749606400, 980995276800, 1961990553600, 25505877196800, 51011754393600, 714164561510400, 1428329123020800

Offset: 0

Author

Per W. Alexandersson, Dec 13 2022

Keywords

• nonn

Comments

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.

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..806
• Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.

Crossrefs

Cf. A016116, A081123.

Programs

Maple

A359039 := proc(n)
    local m ;
    m := floor(n/2) ;
    if type(n,'even') then
        m!*2^m ;
    else
        (m+1)!*2^m ;
    end if;
end proc: # R. J. Mathar, Jul 17 2023
# second Maple program:
a:= n-> ceil(n/2)!*2^floor(n/2):
seq(a(n), n=0..28);  # Alois P. Heinz, Dec 21 2023

Mathematica

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

Formula

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

a(n) = A081123(n+1)*A016116(n). - Alois P. Heinz, Jan 23 2023

Sum_{n>=0} 1/a(n) = 3*sqrt(e) - 2. - Amiram Eldar, Jan 25 2023

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