A393166 Irregular triangular array read by rows. T(n,k) is the total number of inversions over the permutations of [n] whose descent set is contained in S for each subset S of {1,2,...,n-1} where the subsets are ordered first by size and then lexicographically (Cf. A082185).

0, 0, 0, 1, 0, 3, 3, 9, 0, 6, 12, 6, 30, 30, 30, 72, 0, 10, 30, 30, 10, 70, 120, 70, 120, 120, 70, 270, 270, 270, 270, 600, 0, 15, 60, 90, 60, 15, 135, 330, 330, 135, 330, 540, 330, 330, 330, 135, 720, 1170, 720, 1170, 1170, 720, 1170, 1170, 1170, 720, 2520, 2520, 2520, 2520, 2520, 5400

Offset

0,6

References

Richard Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Example 2.2.5.

Links

Table of n, a(n) for n=0..63.

Example

Triangle begins:
  0;
  0;
  0,  1;
  0,  3,  3,  9;
  0,  6, 12,  6, 30, 30, 30,  72;
  0, 10, 30, 30, 10, 70, 120, 70, 120, 120, 70, 270, 270, 270, 270, 600; ...
T(4,7) = 30. S = {2,3}. There are A392945(4,7) = 12 permutations of [4] whose descent set is contained in S: {1, 2, 3, 4}, {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {1, 4, 2, 3}, {1, 4, 3, 2}, {2, 3, 1, 4}, {2, 3, 4, 1}, {2, 4, 1, 3}, {2, 4, 3, 1}, {3, 4, 1, 2}, {3, 4, 2, 1}. There are 0, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5 inversions respectively in these permutations for a total of 30.

Mathematica

steplengths[n_] := Map[Differences, Map[Append[#, n] &, Map[Prepend[#, 0] &, Subsets[Range[n - 1]]]]]; f[list_] := Apply[Times, Map[QFactorial[#, q] &, list]]; numberofinversions[n_, steplengths_] :=Limit[D[QFactorial[n, q]/f[steplengths] // FunctionExpand, q], q -> 1]; Table[Map[numberofinversions[n, #] &, steplengths[n]], {n, 0, 6}]

Crossrefs

Cf. A001809, A392945, A335845, A001809 (last entry in each row), A000217 (second column), A390552, A240796 (row sums).

Sequence in context: A384897 A248241 A109695 * A393447 A372038 A010610

Adjacent sequences: A393163 A393164 A393165 * A393167 A393168 A393169

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 03 2026

Status

approved