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%I #21 Jan 26 2021 16:22:01
%S 1,1,2,4,2,14,8,2,46,62,10,2,282,292,132,12,2,1394,2578,784,268,14,2,
%T 12658,15472,9718,1920,534,16,2,83122,171662,69318,33230,4470,1058,18,
%U 2,985730,1282604,964544,276044,107660,10100,2096,20,2,8012962,17465978,8199268,4851200,1022824,337988,22396,4160,22,2
%N Triangle read by rows: entries give numbers of permutations of [1..n] by absolute 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="/A062867/b062867.txt">Rows n = 0..20, flattened</a>
%F T(n,0) = A062868(n) = A062866(n,0), T(n,k) = 2 * A062866(n,k) for k>0. - _Alois P. Heinz_, Jul 31 2018
%e [1], [2], [4, 2], [14, 8, 2], [46, 62, 10, 2], [282, 292, 132, 12, 2], ...
%e (1,6,2,3,4,5,7) has difference (0,5,-1,-1,-1,-1,0) and signs (0,1,-1,-1,-1,-1,0) with total -3, absolute value is 3. This is one of 268 such permutations of degree 7.
%e Triangle T(n,k) begins:
%e 1;
%e 1;
%e 2;
%e 4, 2;
%e 14, 8, 2;
%e 46, 62, 10, 2;
%e 282, 292, 132, 12, 2;
%e 1394, 2578, 784, 268, 14, 2;
%e 12658, 15472, 9718, 1920, 534, 16, 2;
%e 83122, 171662, 69318, 33230, 4470, 1058, 18, 2;
%e 985730, 1282604, 964544, 276044, 107660, 10100, 2096, 20, 2;
%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)*`if`(i=0, 1, 2), i=0..degree(p)))(b({$1..n}, 0)):
%p seq(T(n), n=0..12); # _Alois P. Heinz_, Jul 31 2018
%t b[s_, t_] := b[s, t] = With[{n = Length[s]}, If[n == 0, x^t, Sum[b[s ~Complement~ {j}, t + Sign[n - j]], {j, s}]]];
%t T[n_] := With[{p = b[Range[n], 0]}, Table[Coefficient[p, x, i]*If[i == 0, 1, 2], {i, 0, Exponent[p, x]}]];
%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 25 2021, after _Alois P. Heinz_ *)
%Y Column k=0 gives A062868.
%Y Row sums give A000142.
%Y Cf. A062866.
%K nice,nonn,tabf
%O 0,3
%A _Olivier Gérard_, Jun 26 2001
%E More terms from _Vladeta Jovovic_, Jun 29 2001
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