Combinatorial correspondences between colored partitions by Dedekind's level 8 partition identities

The study of Dedekind's $\eta$-function and its identities plays a significant role in number theory and combinatorics. In this paper, we study level 8 $\eta$-function identities and their applications to colored partitions. Some of these identities arise from algebraic transformations of known mock theta function expansions. By applying these identities, we deduce combinatorial correspondences between specific classes of colored partitions with prescribed color restrictions. Our work extends existing methods and offers a deeper understanding of the combinatorial properties of partitions, contributing to both theoretical advancements and practical applications in partition theory.