A202363 Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.

1, 6, 3, 36, 24, 12, 240, 180, 120, 60, 1800, 1440, 1080, 720, 360, 15120, 12600, 10080, 7560, 5040, 2520, 141120, 120960, 100800, 80640, 60480, 40320, 20160, 1451520, 1270080, 1088640, 907200, 725760, 544320, 362880, 181440, 16329600, 14515200, 12700800, 10886400, 9072000, 7257600, 5443200, 3628800, 1814400

Offset: 2

Author

Geoffrey Critzer, Jan 09 2013

Keywords

• nonn
• tabl

Comments

Row sums = A001809.

Column for k = 0 is A001286.

Examples

			T(3,1) = 3 because from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have respectively 3 inversion pairs (1,2), (2,3) and (1,3) which are all separated by 1 element.
Triangle T(n,k) begins:
       1;
       6,      3;
      36,     24,     12;
     240,    180,    120,    60;
    1800,   1440,   1080,   720,   360;
   15120,  12600,  10080,  7560,  5040,  2520;
  141120, 120960, 100800, 80640, 60480, 40320, 20160;
  ...
		

Links

• Alois P. Heinz, Rows n = 2..142, flattened

Crossrefs

Cf. A001286, A001809, A055303.

Programs

Mathematica

nn=10;Range[0,nn]!CoefficientList[Series[x^2/2/(1-x)^2/(1-y x),{x,0,nn}],{x,y}]//Grid

Formula

E.g.f.: x^2/2 * (1/(1-x)^2)* (1/(1-y*x)).