A381930 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1,2} such that I(x) + W_0(x)*W_1(x) + W_0(x)*W_2(x) + W_1(x)*W_2(x) = k where I(x) is the number of inversions in x and W_i(x) is the number of occurrences of the letter i in x for i={0,1,2}, n>=0, 0<=k<=floor(2n^2/3).
1, 3, 3, 3, 3, 3, 0, 6, 7, 8, 2, 1, 3, 0, 0, 6, 9, 12, 18, 12, 12, 6, 3, 3, 0, 0, 0, 6, 6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3, 3, 0, 0, 0, 0, 6, 6, 6, 12, 18, 27, 33, 52, 62, 77, 82, 86, 75, 68, 48, 35, 19, 11, 2, 1
Offset: 0
Examples
Triangle T(n,k) begins:
1;
3;
3, 3, 3;
3, 0, 6, 7, 8, 2, 1;
3, 0, 0, 6, 9, 12, 18, 12, 12, 6, 3;
3, 0, 0, 0, 6, 6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3;
...
T(3,3) = 7 because we have: {0, 1, 0}, {0, 1, 2}, {0, 2, 0}, {1, 0, 1}, {1, 2, 1}, {2, 0, 2}, {2, 1, 2}.
Links
- Alois P. Heinz, Rows n = 0..45, flattened
Programs
-
Maple
b:= proc(n, i, j, k) option remember; expand( `if`(n=0, z^(i*j+i*k+j*k), b(n-1, i+1, j, k)*z^(j+k)+ b(n-1, i, j+1, k)*z^k +b(n-1, i, j, k+1))) end: T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)): seq(T(n), n=0..10); # Alois P. Heinz, Mar 10 2025 -
Mathematica
nn = 6; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, q] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^3, {z, 0, nn}],z]] // Grid
Formula
Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^3 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).
From Alois P. Heinz, Mar 10 2025: (Start)
Sum_{k>=0} k * T(n,k) = 9 * A027472(n+1).
Sum_{k>=0} (-1)^k * T(n,k) = A056449(n). (End)
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