Let $p_k(n)$ be a number of ways to express $n$ as a sum of $k$ positive integers. For example $p_2(3)=1$.
Problem 1. Prove that following recurrences are correct:
$p_k(n)=p_{k-1}(n-1)+p_k(n-k)$
$p_k(n)=p_k(n-k)+p_{k-1}(n-k) + ... + p_1(n-k)$
Problem 2. Find $p_2(n)$ and $p_3(n)$.
Unfortunately I was always weak in showing recurrences and I have no idea for 1. I tried to use Ferrers diagrams to see sth but I haven't seen anything. For 2 it is easy to observe that $p_2(n)=\lfloor n/2 \rfloor$ and using first recurrence, that I failed to prove, I got $\displaystyle p_3(n)=\sum_{k=0}^{\lceil n/2 \rceil-1} \left\lfloor\frac{n-3k-1}{2}\right\rfloor$ but I don't know if it's possible to simplify this sum. Maybe some combinatorial interpretation, with Ferrers diagrams for example, will lead to more explicit form for $p_3(n)$?