A062867 - OEIS

%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