I'm looking for a computationally nice enumeration of the $n$-dimensional discrete simplex $$\Delta^n_N = \{ x \in Z^{n} | 0 \leq x_i \leq N \, \text{and} \, x_1 + \cdots x_n = N \}$$
I have an easy way to generate all such points but it does not lead to an easily reversible isomorphism. Since the number of points is $\binom{N+n - 1}{n}$, I need to move to a simple computational representation (that is easily reversible), since for large $N$ the mapping is too large to store in the RAM of a typical computer.
For $n=2$ there is an easy and obvious map which may generalize recursively.