I'm doing a problem and looking for some way to create a specific bijection between $\mathbb{N}$ and my set $W$. $W$ is a set that contains infinite subsets $Y_n$. Each of these subsets contains every subset of $\mathbb{N}$ that adds up to $n$. I already know that each set $Y_n$ contains $\frac{n+ (n \mod 2)}{2}$ many subsets, which means that I basically already have the 2nd index which indexes the sets within $Y_n$. Now the first index needs to follow the following pattern:
| x | n for $Y_n$ |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 3 |
| 5 | 4 |
| 6 | 4 |
| 7 | 5 |
| 8 | 5 |
| 9 | 5 |
It follows this pattern: https://i.stack.imgur.com/BujFv.jpg
Does this function exist at all? If not, what could be some alternatives?