From Dedekind’s Level 12 Identities to Combinatorial Structures of Colored Partitions

The Dedekind η-function plays an important role in number theory, particularly in the study of modular forms, q-series, and partition identities. In this paper, we investigate several level-12 η-function identities and examine their combinatorial implications. These identities are obtained from algebraic transformations of known expansions involving mock theta functions, which were originally introduced by Srinivasa Ramanujan. By employing classical q-series techniques and modular transformations, we derive identities that reveal interesting relationships among η-functions. We further interpret these identities combinatorially to establish correspondences between specific classes of colored partitions with prescribed color restrictions. These results provide new insights into the structure of colored partition functions and highlight the interplay between mock theta functions, Dedekind η-function identities, and combinatorial partition theory. Our findings contribute to a deeper understanding of the connections between modular forms and colored partitions and suggest further directions for research in number theory and combinatorics.