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%I #25 Jul 31 2018 16:55:47
%S 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,
%T 7,134,392,1289,1394,1289,392,134,7,1,1,8,267,960,4859,7736,12658,
%U 7736,4859,960,267,8,1,1,9,529,2235,16615,34659,85831,83122,85831,34659,16615,2235,529,9,1
%N Triangle of number of permutations by barycenter.
%C The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects.
%H Alois P. Heinz, <a href="/A062866/b062866.txt">Rows n = 0..20, flattened</a>
%F From _Alois P. Heinz_, Jul 31 2018: (Start)
%F T(n,k) = T(n,-k).
%F Sum_{k>=0} T(n,k) = A179566(n). (End)
%e (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.
%e Triangle begins:
%e : 1 ;
%e : 1 ;
%e : 2 ;
%e : 1, 4, 1 ;
%e : 1, 4, 14, 4, 1 ;
%e : 1, 5, 31, 46, 31, 5, 1 ;
%e : 1, 6, 66, 146, 282, 146, 66, 6, 1 ;
%e : 1, 7, 134, 392, 1289, 1394, 1289, 392, 134, 7, 1 ;
%e : 1, 8, 267, 960, 4859, 7736, 12658, 7736, 4859, 960, 267, 8, 1 ;
%p b:= proc(s, t) option remember; (n-> `if`(n=0, x^t,
%p add(b(s minus {j}, t+signum(n-j)), j=s)))(nops(s))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b({$1..n}, 0)):
%p seq(T(n), n=0..11); # _Alois P. Heinz_, Jul 31 2018
%t row[n_] := Sort[Tally[Total[Sign[# - Range[n]]]& /@ Permutations[Range[n]] ]][[All, 2]]; Array[row, 9] // Flatten (* _Jean-François Alcover_, Oct 07 2016 *)
%Y Columns k=0-4 give: A062868, A179562, A169934, A179564, A179565.
%Y Row sums give A000142.
%Y Cf. A062867, A179566.
%K nice,nonn,tabf
%O 0,3
%A _Olivier Gérard_, Jun 26 2001
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