A380993 Irregular triangular array read by rows. T(n,k) is the number of ternary words of length n containing at least one copy of each letter and having exactly k inversions, n>=3, 0<=k<=floor(n^2/3).

1, 2, 2, 1, 3, 6, 9, 9, 6, 3, 6, 12, 21, 27, 30, 24, 18, 9, 3, 10, 20, 38, 55, 74, 81, 80, 69, 53, 34, 17, 8, 1, 15, 30, 60, 93, 138, 174, 210, 216, 219, 195, 165, 120, 84, 48, 27, 9, 3, 21, 42, 87, 141, 222, 303, 405, 480, 546, 579, 588, 552, 498, 414, 324, 240, 162, 99, 54, 27, 9, 3

Offset: 3

Author

Geoffrey Critzer, Feb 11 2025

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
   1,  2,  2,  1;
   3,  6,  9,  9,  6,  3;
   6, 12, 21, 27, 30, 24, 18,  9,  3;
  10, 20, 38, 55, 74, 81, 80, 69, 53, 34, 17, 8, 1;
  ...
T(4,2) = 9 because we have: {0, 1, 2, 0}, {0, 2, 0, 1}, {0, 2, 1, 1}, {0, 2, 2, 1}, {1, 0, 0, 2}, {1, 0, 2, 1}, {1, 1, 0, 2}, {1, 2, 0, 2}, {2, 0, 1, 2}.
		

Links

• Alois P. Heinz, Rows n = 3..50, flattened

Crossrefs

Cf. A056454, A129529, A001117 (row sums).

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, `if`(nops(subs(0=
      [][], l))=3, 1, 0), add(expand(x^([0, l[1], l[1]+l[2]][j])*
      b(n-1, subsop(j=`if`(j=3, 1, l[j]+1), l))), j=1..3))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(T(n), n=3..10);  # Alois P. Heinz, Feb 12 2025

Mathematica

nn = 8; B[n_] := FunctionExpand[QFactorial[n, u]];
e[z_] := Sum[z^n/B[n], {n, 0, nn}];
Drop[Map[CoefficientList[#, u] &,
   Map[Normal[Series[#, {u, 0, Binomial[nn, 2]}]] &,
    Table[B[n], {n, 0, nn}] CoefficientList[
      Series[(e[z] - 1)^3, {z, 0, nn}], z]]], 3] // Grid

Formula

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/B(n) = (e(x)-1)^3 where B(n) = Product_{i=1..n} (q^i-1)/(q-1) and e(x) = Sum_{n>=0} x^n/B(n).

Sum_{k=0..floor(n^2/3)} (-1)^k * T(n,k) = A056454(n). - Alois P. Heinz, Feb 12 2025