Some Inplace Identities for Integer Compositions

In this paper, we give two new identities for compositions, or ordered partitions, of integers. These two identities are based on closely-related integer partition functions which have recently been studied. Thanks to the structure inherent in integer compositions, we are also able to extensively generalize both of these identities. Bijective proofs are given and generating functions are provided for each of the types of compositions which arise. A number of arithmetic properties satisfied by the functions which count such compositions are also highlighted.