A062867 Triangle read by rows: entries give numbers of permutations of [1..n] by absolute barycenter.

1, 1, 2, 4, 2, 14, 8, 2, 46, 62, 10, 2, 282, 292, 132, 12, 2, 1394, 2578, 784, 268, 14, 2, 12658, 15472, 9718, 1920, 534, 16, 2, 83122, 171662, 69318, 33230, 4470, 1058, 18, 2, 985730, 1282604, 964544, 276044, 107660, 10100, 2096, 20, 2, 8012962, 17465978, 8199268, 4851200, 1022824, 337988, 22396, 4160, 22, 2

Offset: 0

Author

Olivier Gérard, Jun 26 2001

Keywords

• nice
• nonn
• tabf

Comments

The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects.

Examples

			[1], [2], [4, 2], [14, 8, 2], [46, 62, 10, 2], [282, 292, 132, 12, 2], ...
(1,6,2,3,4,5,7) has difference (0,5,-1,-1,-1,-1,0) and signs (0,1,-1,-1,-1,-1,0) with total -3, absolute value is 3. This is one of 268 such permutations of degree 7.
Triangle T(n,k) begins:
       1;
       1;
       2;
       4,       2;
      14,       8,      2;
      46,      62,     10,      2;
     282,     292,    132,     12,      2;
    1394,    2578,    784,    268,     14,     2;
   12658,   15472,   9718,   1920,    534,    16,    2;
   83122,  171662,  69318,  33230,   4470,  1058,   18,  2;
  985730, 1282604, 964544, 276044, 107660, 10100, 2096, 20, 2;
		

Links

• Alois P. Heinz, Rows n = 0..20, flattened

Crossrefs

Column k=0 gives A062868.

Row sums give A000142.

Cf. A062866.

Programs

Maple

b:= proc(s, t) option remember; (n-> `if`(n=0, x^t,
      add(b(s minus {j}, t+signum(n-j)), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)*`if`(i=0, 1, 2), i=0..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jul 31 2018

Mathematica

b[s_, t_] := b[s, t] = With[{n = Length[s]}, If[n == 0, x^t, Sum[b[s  ~Complement~ {j}, t + Sign[n - j]], {j, s}]]];
T[n_] := With[{p = b[Range[n], 0]}, Table[Coefficient[p, x, i]*If[i == 0, 1, 2], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2021, after Alois P. Heinz *)

Formula

T(n,0) = A062868(n) = A062866(n,0), T(n,k) = 2 * A062866(n,k) for k>0. - Alois P. Heinz, Jul 31 2018

Extensions

More terms from Vladeta Jovovic, Jun 29 2001