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
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
KEYWORD
nice,nonn,tabf
AUTHOR
Olivier Gérard, Jun 26 2001
STATUS
approved