As remarked on the main partitions page, the proofs of some partition identities derive from manipulating the Ferrers graph of a partition.

For example, if the Ferrers graph of a partition \(\lambda\) of a positive integer \(n\) is reflected across the main diagonal, a new partition of \(n\), denoted by \(\lambda'\) and called the conjugate of \(\lambda\), is obtained (see the page Conjugate and Self-conjugate Partitions for more on this topic).
  As an example, the conjugate of the partition \(5+ 4+ 4+ 2+ 1\) is the partition \(5+4+ 3+ 3+ 1\)- see Figure 2.

  It is not difficult to see that the conjugate of a partition into distinct parts is a partition in which every part from 1 to the largest part occurs at least once, and vice versa, thus leading to a bijective proof the following partition identity:

Theorem For each positive integer \(n\), let \(p_{\mathcal{D}}(n)\) denote the number of partitions of \(n\) into distinct parts, and let \(p_{\mathcal{L}}(n)\) denote the number of partitions of \(n\) in which every part from 1 to the largest part occurs at least once. Then \[ p_{\mathcal{D}}(n)=p_{\mathcal{L}}(n). \]

As a second example of a combinatorial proof of a partition identity, consider the Ferrers graph of a partition into distinct odd parts. If the each of the rows (each corresponding to a distinct odd part) is bent at right-angles around the central node, and these right-angled shapes are nested together to form the Ferrers graph of a new partition, then this new partition is a self-conjugate partition (one that is equal to its own conjugate). This process is reversible.

Further, noting that the size of the Durfee square of the self-conjugate partition equals the number of parts in the partition into distinct odd parts, what has been shown is that for any positive integers \(m\) and \(n\),

# self-conjugate partitions of n with Durfee square of side m = # of partitions of n into m distinct odd parts.

Other pairs of restricted partition counting functions can also be shown to take the same value at each positive integer \(n\), by similarly finding bijections between the sets of partitions counted by the functions in each pair.

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