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

2,3

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

Links

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

Example

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;
  ...

Mathematica

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

Crossrefs

Cf. A000166, A000387, A003221, A216239.

Sequence in context: A182138 A258123 A121583 * A246862 A338773 A194686

Adjacent sequences: A228921 A228922 A228923 * A228925 A228926 A228927

Keyword

nonn,tabf

Author

Geoffrey Critzer, Sep 08 2013

Status

approved