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

0,3

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.

Links

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

Formula

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

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

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

Example

(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  ;

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 *)

Crossrefs

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

Row sums give A000142.

Cf. A062867, A179566.

Sequence in context: A255878 A162944 A140582 * A131035 A339833 A118745

Adjacent sequences: A062863 A062864 A062865 * A062867 A062868 A062869

Keyword

nice,nonn,tabf

Author

Olivier Gérard, Jun 26 2001

Status

approved