A Combinatorial Proof of Partition Identities for the Number of Distinct Part Sizes

In 1984, Bressoud and Subbarao established an interesting weighted partition identity connecting the partition function and the divisor function. Recently, the first author, jointly with Bhoria, Eyyunni, and Maji, extended the work of Bressoud and Subbarao and obtained the following analogous weighted partition identity. For any $n\in \mathbb{N}$, they obtained \begin{equation*} p^{(2)}(n) = \sum_{π\in $\mathcal{D}$(n)} (-1)^{ # (π) }s(π) \left( \ell(π) - s(π) \right). \end{equation*} In the present paper, we provide a combinatorial proof of the above identity and extend the result for $p^{(k)}(n)$, where $p^{(k)}(n)$ denotes the number of partitions of $n$ into exactly $k$ distinct part sizes.