A062866 Triangle of number of permutations by barycenter.

1, 1, 2, 1, 4, 1, 1, 4, 14, 4, 1, 1, 5, 31, 46, 31, 5, 1, 1, 6, 66, 146, 282, 146, 66, 6, 1, 1, 7, 134, 392, 1289, 1394, 1289, 392, 134, 7, 1, 1, 8, 267, 960, 4859, 7736, 12658, 7736, 4859, 960, 267, 8, 1, 1, 9, 529, 2235, 16615, 34659, 85831, 83122, 85831, 34659, 16615, 2235, 529, 9, 1

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

			(1,3,2,5,7,6,4) has difference (0,1,-1,1,2,0,-3) and signs (0,1,-1,1,1,0,-1) with total 1. This is one of 1289 such permutations of degree 7.
Triangle begins:
:                                  1                              ;
:                                  1                              ;
:                                  2                              ;
:                           1,     4,    1                        ;
:                     1,    4,    14,    4,    1                  ;
:               1,    5,   31,    46,   31,    5,   1             ;
:          1,   6,   66,  146,   282,  146,   66,   6,   1        ;
:     1,   7, 134,  392, 1289,  1394, 1289,  392, 134,   7, 1     ;
:  1, 8, 267, 960, 4859, 7736, 12658, 7736, 4859, 960, 267, 8, 1  ;
		

Links

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

Crossrefs

Columns k=0-4 give: A062868, A179562, A169934, A179564, A179565.

Row sums give A000142.

Cf. A062867, A179566.

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), i=ldegree(p)..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..11);  # Alois P. Heinz, Jul 31 2018

Mathematica

row[n_] := Sort[Tally[Total[Sign[# - Range[n]]]& /@ Permutations[Range[n]] ]][[All, 2]]; Array[row, 9] // Flatten (* Jean-François Alcover, Oct 07 2016 *)

Formula

From Alois P. Heinz, Jul 31 2018: (Start)

T(n,k) = T(n,-k).

Sum_{k>=0} T(n,k) = A179566(n). (End)

Conjecture: e.g.f.: Sum_{n>=0} Sum_{k} T(n,k) * t^k * z^n / n! = (1-t^2) * exp(z) / (exp(t*z) - t^2 * exp(z/t)). - Robert S. Maier, Jan 17 2025