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A062866 Triangle of number of permutations by barycenter.

%I A062866 #31 Jan 19 2025 22:39:08
%S A062866 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 A062866 7,134,392,1289,1394,1289,392,134,7,1,1,8,267,960,4859,7736,12658,
%U A062866 7736,4859,960,267,8,1,1,9,529,2235,16615,34659,85831,83122,85831,34659,16615,2235,529,9,1
%N A062866 Triangle of number of permutations by barycenter.
%C A062866 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 A062866 Alois P. Heinz, <a href="/A062866/b062866.txt">Rows n = 0..20, flattened</a>
%F A062866 From _Alois P. Heinz_, Jul 31 2018: (Start)
%F A062866 T(n,k) = T(n,-k).
%F A062866 Sum_{k>=0} T(n,k) = A179566(n). (End)
%F A062866 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
%e A062866 (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 A062866 Triangle begins:
%e A062866 :                                  1                              ;
%e A062866 :                                  1                              ;
%e A062866 :                                  2                              ;
%e A062866 :                           1,     4,    1                        ;
%e A062866 :                     1,    4,    14,    4,    1                  ;
%e A062866 :               1,    5,   31,    46,   31,    5,   1             ;
%e A062866 :          1,   6,   66,  146,   282,  146,   66,   6,   1        ;
%e A062866 :     1,   7, 134,  392, 1289,  1394, 1289,  392, 134,   7, 1     ;
%e A062866 :  1, 8, 267, 960, 4859, 7736, 12658, 7736, 4859, 960, 267, 8, 1  ;
%p A062866 b:= proc(s, t) option remember; (n-> `if`(n=0, x^t,
%p A062866       add(b(s minus {j}, t+signum(n-j)), j=s)))(nops(s))
%p A062866     end:
%p A062866 T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b({$1..n}, 0)):
%p A062866 seq(T(n), n=0..11);  # _Alois P. Heinz_, Jul 31 2018
%t A062866 row[n_] := Sort[Tally[Total[Sign[# - Range[n]]]& /@ Permutations[Range[n]] ]][[All, 2]]; Array[row, 9] // Flatten (* _Jean-François Alcover_, Oct 07 2016 *)
%Y A062866 Columns k=0-4 give: A062868, A179562, A169934, A179564, A179565.
%Y A062866 Row sums give A000142.
%Y A062866 Cf. A062867, A179566.
%K A062866 nice,nonn,tabf
%O A062866 0,3
%A A062866 _Olivier Gérard_, Jun 26 2001