Combinatorial proofs of some enumeration identities (Q1109775)

The author first gives a proof by generating function, as also a combinatorial proof of the following result: Theorem. The number of compositions of n with exactly m parts equals the number of partitions into m distinct parts with largest part n. The above theorem and some related results are then used in proving the following theorem combinatorially: Theorem. The number of compositions of n is the same as the number of self-conjugate partitions with largest part equal to n. Examples are cited to explain the meaning of the results.