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.
  For example, the conjugate of the partition \(5+ 4+ 4+ 2+ 1\) is the partition \(5+4+ 3+ 3+ 1\)- see Figure 2.

If the Ferrers graph of the conjugate of a partition is the same as the Ferrers graph of the original partition, then the partition is said to be a self-conjugate partition.
For example, it can be easily seen from the Ferrers graph that the partition \(6+ 5+ 3+ 2+ 2+ 1 \) is a self-conjugate partitio n of 19.

The number of self-conjugate partitions of any positive integer \(n\) are equinumerous with the number of partitions of \(n\) into distinct odd parts (see the Combinatorial Proofs of Partition Identities page for a proof of this).

Conjugate partitions may also be used to prove that, for integers \(a,b\) and \(c\) with \(a>b>1, a>c>1\), the number of partitions of \(a−c \) into exactly \(b−1\) parts, none exceeding \(c\), equals the number of partitions of \(a − b\) into exactly \(c − 1\) parts, none exceeding \(b\) (the reader may try to prove this using Ferrers graphs).

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