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
T(n,0) = A062868(n) = A062866(n,0), T(n,k) = 2 * A062866(n,k) for k>0. - Alois P. Heinz, Jul 31 2018
EXAMPLE
[1], [2], [4, 2], [14, 8, 2], [46, 62, 10, 2], [282, 292, 132, 12, 2], ...
(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.
Triangle T(n,k) begins:
1;
1;
2;
4, 2;
14, 8, 2;
46, 62, 10, 2;
282, 292, 132, 12, 2;
1394, 2578, 784, 268, 14, 2;
12658, 15472, 9718, 1920, 534, 16, 2;
83122, 171662, 69318, 33230, 4470, 1058, 18, 2;
985730, 1282604, 964544, 276044, 107660, 10100, 2096, 20, 2;
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)*`if`(i=0, 1, 2), i=0..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..12); # Alois P. Heinz, Jul 31 2018
MATHEMATICA
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[n_] := With[{p = b[Range[n], 0]}, Table[Coefficient[p, x, i]*If[i == 0, 1, 2], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nice,nonn,tabf
AUTHOR
Olivier Gérard, Jun 26 2001
EXTENSIONS
More terms from Vladeta Jovovic, Jun 29 2001
STATUS
approved