For a given partition $T$ of a positive integer $n$, let $g_{T,n}$ denote the number of distinct parts in $T$ and let $h_{T,n}$ denote the number of $1$'s appearing as parts in $T$. Let $$p_1(n) = \sum \limits_{T} h_{T,n}$$ and $$p_{p} (n) = \sum \limits_{T} g_{T,n}.$$ Where $T$ varies over all the partitions of $n$.
Show that, for any positive integer $n$, $p_{1} (n) = p_{p} (n)$.
How do I proceed to prove it? Please help me in this regard.
Thank you very much.
EDIT $:$
I have found that $p_{1} (n) = \sum \limits_{k=1}^{n-1} P(k)$ where $P(k)$ is the partition function evaluated at $k$.
As you say, $p_1(n)=\sum_{k=0}^{n-1}P(k)$. I'll show the same holds for $p_p(n)$.
Now $p_p(n)=q_1(n)+q_2(n)+\cdots+q_n(n)$ where $q_k(n)$ is the number of partitions of $n$ having $k$ as a part. But removing $k$ as a part from a partition of $n$ having $k$ as a part yields a partition of $n-k$. Each partition of $n-k$ arises exactly once in this way. Therefore, $q_k(n)=P(n-k)$, and $$p_p(n)=\sum_{k=1}^n P(n-k)=\sum_{k=0}^{n-1}P(k).$$