A335873 Total number of points in all permutations of [n] that are fixed or reflected.

0, 1, 4, 10, 48, 216, 1440, 9360, 80640, 685440, 7257600, 76204800, 958003200, 11975040000, 174356582400, 2528170444800, 41845579776000, 690452066304000, 12804747411456000, 236887827111936000, 4865804016353280000, 99748982335242240000, 2248001455555215360000

Offset

0,3

Comments

A permutation p of [n] has fixed point j if p(j) = j, it has reflected point j if p(n+1-j) = j. A point can be fixed and reflected at the same time.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)

Wikipedia, Permutation

Formula

E.g.f.: 2*x/(1-x) - (log(1+x) - log(1-x))/2.

a(0) = 0, a(n) = 2*n! - (n mod 2)*(n-1)! for n > 0.

a(n) = (n-1)*(4*a(n-1)+(n-2)*(4*n-3)*a(n-2))/(4*n-7) for n >= 2, a(n) = n for n < 2.

a(n) = Sum_{k=1..n} k * A335872(n,k).

Example

a(3) = 10: (1)(2)(3), (1)32, 21(3), 23(1), (3)12, (3)(2)(1).

Maple

b:= proc(s, i) option remember; (n-> `if`(n=0, [1, 0],
      add((p-> p+[0, `if`(j in {i, n}, p[1], 0)])(
        b(s minus {j}, i+1)), j=s)))(nops(s))
    end:
a:= n-> b({$1..n}, 1)[2]:
seq(a(n), n=0..14);
# second Maple program:
a:= n-> `if`(n=0, 0, 2*n! -`if`(n::odd, (n-1)!, 0)):
seq(a(n), n=0..22);
# third Maple program:
a:= proc(n) option remember; `if`(n<2, n, (n-1)*
      (4*a(n-1)+(n-2)*(4*n-3)*a(n-2))/(4*n-7))
    end:
seq(a(n), n=0..22);

Mathematica

a[n_] :=  If[n == 0, 0, 2 n! - If[OddQ[n], (n-1)!, 0]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Aug 24 2021, from 2nd Maple program *)

Crossrefs

Bisection (even part) gives 2 * A010050(n) for n>0.

Cf. A000142, A005359, A306258, A335872.

Sequence in context: A197664 A099606 A149231 * A364281 A173086 A081565

Adjacent sequences: A335870 A335871 A335872 * A335874 A335875 A335876

Keyword

nonn,easy

Author

Alois P. Heinz, Jun 28 2020

Status

approved