Group theory provides a systematic way of thinking about symmetry. Wreath products are group-theoretic constructions that have been used to model symmetries in a whole variety of research disciplines. The goal of this paper is to calculate the partition functions induced by these algebraic structures when organized iteratively under the symmetries imposed by the permutation and cyclic groups. The resulting hierarchical structures are modeled as Cayley-like trees from a statistical mechanics

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... t of view, whereas the interactions between the nodes of those trees are defined by the actions induced by these groups. The emphasis is put on the analytic combinatorics treatment of the problem as a way to obtain closed expressions for the partition functions. Furthermore, the advantages of the singularity analysis and symbolic techniques of this mathematical theory are stressed as a way of extracting asymptotic information and setting up functional relations between partition functions.