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A062866
Triangle of number of permutations by barycenter.
10
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
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.
Sequence in context: A255878 A162944 A140582 * A131035 A339833 A118745
KEYWORD
nice,nonn,tabf
AUTHOR
Olivier Gérard, Jun 26 2001
STATUS
approved