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A335872 Number T(n,k) of permutations of [n] having k points that are fixed or reflected; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 3
1, 0, 1, 0, 0, 2, 0, 4, 0, 2, 4, 0, 16, 0, 4, 16, 36, 32, 32, 0, 4, 80, 192, 216, 128, 96, 0, 8, 672, 1472, 1440, 984, 320, 144, 0, 8, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 48768, 103568, 104448, 65920, 28544, 9312, 1792, 512, 0, 16 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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, Rows n = 0..22, flattened

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

Wikipedia, Permutation

FORMULA

Sum_{k=1..n} k * T(n,k) = A335873(n).

T(n,n-2) = floor((n-1)^2/2) * 2^floor(n/2).

EXAMPLE

      1;

      0,      1;

      0,      0,      2;

      0,      4,      0,     2;

      4,      0,     16,     0,     4;

     16,     36,     32,    32,     0,    4;

     80,    192,    216,   128,    96,    0,    8;

    672,   1472,   1440,   984,   320,  144,    0,   8;

   4752,  10752,  11776,  7680,  3936, 1024,  384,   0, 16;

  48768, 103568, 104448, 65920, 28544, 9312, 1792, 512,  0, 16;

  ...

MAPLE

b:= proc(s, i, t) option remember; (n-> `if`(n=0, x^t, add(

      b(s minus {j}, i+1, t+`if`(j in {i, n}, 1, 0)), j=s)))(nops(s))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n}, 1, 0)):

seq(T(n), n=0..10);

MATHEMATICA

b[s_, i_, t_] := b[s, i, t] = With[{n = Length[s]}, If[n == 0, x^t, Sum[b[s ~Complement~ {j}, i+1, t + If[j == i || j == n, 1, 0]], {j, s}]]];

T[n_] := CoefficientList[b[Range[n], 1, 0], x];

T /@ Range[0, 10] // Flatten (* Jean-Fran├žois Alcover, Feb 13 2021, after Alois P. Heinz *)

CROSSREFS

Column k=0 gives A003471.

Main diagonal gives A016116.

Row sums give A000142.

Cf. A008290, A335873.

Sequence in context: A326722 A279228 A181481 * A239489 A259759 A119607

Adjacent sequences:  A335869 A335870 A335871 * A335873 A335874 A335875

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 28 2020

STATUS

approved

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Last modified May 7 13:12 EDT 2021. Contains 343650 sequences. (Running on oeis4.)