Let $n,p \in \mathbb N$ and consider the integer partition of $$\left\lfloor \frac{n(p-1)}{2} \right\rfloor$$ into $p$ or less parts, each of which is less or equal to $n-1$.

Can the number of partitions be bounded by $$ {m+p-1 \choose p}$$ with a suitable $m < n-1$?