Durfee-square decomposition of the partition series

Let $\lambda$ be a partition with Durfee length $k$. Removing its $k\times k$ Durfee square leaves two pieces: a right-hand piece $\alpha$ consisting of the cells to the right of the square, and a lower piece $\beta$ consisting of the cells below the square. The piece $\alpha$ has at most $k$ rows, while each row of $\beta$ has length at most $k$. Conversely, given $k$, a partition $\alpha$ with at most $k$ parts, and a partition $\beta$ with all parts at most $k$, one reconstructs $\lambda$ uniquely by adjoining $\alpha$ to the right side and $\beta$ below the square.

For fixed $k$, the square contributes the factor $x^{k^2}$. By [L2], the right-hand piece $\alpha$ has the same generating function as partitions with parts at most $k$, namely ${\prod }_{i=1}^k(1-x^i{)}^{-1}$; the lower piece $\beta$ has the same generating function for the same direct multiplicity reason. Thus partitions whose Durfee square has size $k$ contribute $x^{k^2}\mathrm{/}{\prod }_{i=1}^k(1-x^i{)}^2$.

Every partition has exactly one Durfee length, so summing the contributions of step 2.1 over all $k\ge 0$ counts every partition exactly once. Therefore the coefficient of $x^n$ on the right is $p(n)$ for every $n\ge 0$, and [L1] gives the displayed identity.