Does anyone know a closed-form expression for a bijection between $\mathbb{N}^k$ and $\mathbb{N}$? I want to publish an article and one of its results is a simple closed-form expression for a natural bijection between $\mathbb{N}^k$ and $\mathbb{N}$. I wish to know whether it is already known.

Unfortunately, my manuscript was rejected by the American Mathematical Monthly. They said the following: 
"The paper concentrates on the explicit construction of a bijection between $\bf N\times N$ and $\bf N$ and similar bijections for ${\bf N}^k$. However, for me it is quite unclear what is the intended audience and what is the main point of the paper. An explicit polynomial formula for the bijection indeed is nice, but it is definitely not new (I have seen it in logical textbooks). If the main point is a nice accessible explanation (for a wide audience) why this formula is true, it is possible, but the style of the paper with a lot of formulas is quite confusing and is not suitable for this goal.
So I am sorry to say that IMHO this paper does not look suitable for AMM (and, probably, for other journals I can think of)..."
So, as I have promised, I came back here to present you my formula:
$$\Psi_k(n_1, \dots, n_k) = \sum_{i=1}^{k}\binom{i-1+n_1+\dots +n_i}{i}$$
I will still try to present my work in some colloquium, before revealing how I got this formula.
