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A062867 Triangle read by rows: entries give numbers of permutations of [1..n] by absolute barycenter. 5
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, 8012962, 17465978, 8199268, 4851200, 1022824, 337988, 22396, 4160, 22, 2 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Column k=0 gives A062868.

Cf. A062866.

Sequence in context: A006018 A152666 A153801 * A264027 A113539 A215055

Adjacent sequences:  A062864 A062865 A062866 * A062868 A062869 A062870

KEYWORD

nice,nonn,tabf

AUTHOR

Olivier Gérard, Jun 26 2001

EXTENSIONS

More terms from Vladeta Jovovic, Jun 29 2001

STATUS

approved

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Last modified December 6 21:45 EST 2019. Contains 329809 sequences. (Running on oeis4.)