Counting Summations

It is possible to write five as a sum in exactly six different ways:

\begin{aligned} 4 + 1 \\ 3 + 2 \\ 3 + 1 + 1 \\ 2 + 2 + 1 \\ 2 + 1 + 1 + 1 \\ 1 + 1 + 1 + 1 + 1 \end{aligned}

How many different ways can one hundred be written as a sum of at least two positive integers?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.