A228924 Irregular triangular array read by rows: T(n,k) is the number of derangement permutations of [n] that have exactly k inversions; n>=2, 1<=k<=binomial(n,2) for even n, 1<=k<=binomial(n,2)-1 for odd n.

1, 0, 2, 0, 1, 4, 1, 2, 1, 0, 0, 4, 8, 4, 10, 10, 6, 2, 0, 0, 1, 12, 18, 16, 35, 44, 47, 40, 25, 14, 8, 4, 1, 0, 0, 0, 6, 32, 44, 60, 118, 160, 208, 244, 244, 214, 174, 140, 104, 64, 30, 10, 2, 0, 0, 0, 1, 24, 83, 118, 206, 388, 565, 802, 1068, 1308, 1466, 1508, 1479, 1414, 1290, 1076, 806, 544, 333, 186, 96, 46, 19, 6, 1

Offset: 2

Author

Geoffrey Critzer, Sep 08 2013

Keywords

• nonn
• tabf

Comments

Row sums = A000166.

Sum_{k>=1} T(n,k)*k = A216239(n).

Sum_{even k} T(n,k) = A003221(n) and Sum_{odd k} T(n,k) = A000387(n).

It would be nice to have a closed formula for T(n,k). - Alois P. Heinz, Dec 31 2014

Examples

			Triangle T(n,k) begins:
  1;
  0, 2;
  0, 1, 4,  1,  2,  1;
  0, 0, 4,  8,  4, 10, 10,  6,  2;
  0, 0, 1, 12, 18, 16, 35, 44, 47, 40, 25, 14, 8, 4, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 2..23, flattened

Crossrefs

Cf. A000166, A000387, A003221, A008302, A216239.

Programs

Mathematica

Map[Distribution[#,Range[1,Max[#]]]&,Table[Map[Inversions,Derangements[n]],{n,2,6}]]//Grid