A337126 Irregular triangular array read by rows. T(n,k) is the number of permutations of {1,2,...,n} with descent set {1,3,5,...,m} (where m is the greatest odd integer less than n) that have exactly k inversions, n=0, k=0, or n>0, 0<=k<=ceiling((n-1)^2/2).

1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 0, 1, 2, 5, 7, 9, 10, 10, 8, 5, 3, 1, 0, 0, 0, 1, 3, 7, 13, 19, 26, 32, 35, 35, 32, 26, 19, 13, 7, 3, 1, 0, 0, 0, 0, 1, 3, 9, 18, 32, 50, 72, 95, 117, 134, 143, 145, 138, 122, 101, 78, 55, 36, 21, 10, 4, 1

Offset: 0

Author

Geoffrey Critzer, Aug 17 2020

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
  1;
  1;
  0, 1;
  0, 1, 1;
  0, 0, 1, 1, 2, 1;
  0, 0, 1, 2, 3, 4, 3, 2, 1;
  0, 0, 0, 1, 2, 5, 7, 9, 10, 10, 8, 5, 3, 1;
  ...
T(6,5) = 5 because we have: {2, 1, 5, 4, 6, 3}, {2, 1, 6, 3, 5, 4},
  {3, 1, 5, 2, 6, 4}, {3, 2, 4, 1, 6, 5}, {4, 1, 3, 2, 6, 5}.
		

References

• R. Stanley, Enumerative Combinatorics, volume 1, second edition, Cambridge University Press (2012), p.295.

Links

• Alois P. Heinz, Rows n = 0..50, flattened

Crossrefs

Cf. A000111 (row sums), A337193 (total number of inversions).

Programs

Maple

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1, add(
      x^`if`(t=0, o-1+j, u-j)*b(o-1+j, u-j, 1-t), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 17 2020

Mathematica

Table[a = Drop[Subsets[Table[i, {i, 1, n - 1, 2}]], 1];f[list_] := (-1)^(Floor[n/2] - Length[list]) QBinomial[n, list[[1]], q] Product[
     QBinomial[n - list[[i]], list[[i + 1]] - list[[i]], q], {i, 1,
      Length[list] - 1}]; CoefficientList[Expand[FunctionExpand[Total[Map[f, a]] + (-1)^(Floor[n/2])]], q], {n, 0, 8}] // Grid
(* Second program: *)
b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1, Sum[x^If[t == 0, o - 1 + j, u - j]*b[o - 1 + j, u - j, 1 - t], {j, 1, u}]]];
T[n_] := CoefficientList[b[n, 0, 0], x];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Formula

Sum_{k=1..ceiling((n-1)^2/2)} k * T(n,k) = A337193(n).