Block-Separated Overpartitions: Fibonacci Structure and Euler Factorization

Abstract

We introduce and study block-separated overpartitions, a constrained family of overpartitions in which no two consecutive distinct part-blocks are both overlined. This local restriction produces a new sequence that naturally interpolates between classical partitions and unrestricted overpartitions. We show that the internal decoration of distinct part-blocks is governed by Fibonacci-type combinatorics: once the set of distinct part-sizes is fixed, the admissible overlining patterns are counted by Fibonacci numbers. This leads to a symmetric-function expansion of the generating function and a two-state transfer-matrix formulation. After extracting the Euler product, we obtain normalized recurrences, second-order scalar recurrences, determinantal representations, and a continued-fraction description of finite truncations. Finally, we determine the asymptotic growth of the counting function, and prove that block-separated overpartitions share the same exponential scale as ordinary partitions, with a modified subexponential constant.