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

0,3

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.

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.

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

Example

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

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)]

Crossrefs

Cf. A016116, A081123.

Sequence in context: A286866 A334764 A218861 * A078222 A078223 A291405

Adjacent sequences: A359036 A359037 A359038 * A359040 A359041 A359042

Keyword

nonn

Author

Per W. Alexandersson, Dec 13 2022

Status

approved