How can I construct a Smolyak index Matrix?

Hello everybody.

I want to program a Smolyak matrix of indices, as in this article. For that, I need to decide the number of dimensions, d, and the level, l. The matrix has as many rows as dimensions and the sum of all elements on a row has to be between the values of d and d+l; also, the number formed by the elements of the row has to be as high as low as possible.

These are some examples of what I want:

Example 1: d=2, l=1.

In this case, the index matrix will look like

1 1

1 2

2 1

The first row is equal to d and the second and third are equal to d+l. Also, the number obtained from adding the row is as small as possible. Se 11, 12 and 21.

If the number of dimensions is 3 instead, the matrix would look like

1 1 1

1 1 2

1 2 1

2 1 1

If d=2 and also l=2 the matrix will look like

1 1

1 2

2 1

1 3

2 2

3 1

If someone has programmed how to construct matrices like this, I would be extremely thankful if they shared their code.

Thanks for reading.

Regards.

Jaime.