In this research, we explore partition functions related to identities of Euler type and partitions with initial repetitions. On identities of Euler type, we mainly focus on identities due to P. A. MacMahon, G. E. Andrews and M. V. Subbarao. We give new bijections for partition theorems due to MacMahon and Andrews and provide a generalization of Subbarao’s finitization theorem. For the original Euler’s identity which gave rise to identities of Euler type, Andrews and M. Merca found a new combinatorial interpretation for the total number of even parts in all partitions of n into distinct parts. We genaralize their result and establish more variations with connections to some of the work of S. Fu and D. Tang. By conjugating partitions into distinct parts, we obtain partitions without gaps. Andrews extended the notion of partitions without gaps to partitions with initial repetitions. We study partitions with initial repetitions and give several Legendre theorems. These theorems provide partition-theoretic interpretations of some well known q-series identities of Rogers-Ramanujan type. We further deduce parity formulas for partition functions associated with this class of partitions.
