Block-Separated Overpartitions: Fibonacci Structure and Euler Factorization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911491608805376 |
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| author | Mehiri, El-Mehdi |
| author_facet | Mehiri, El-Mehdi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block-Separated Overpartitions: Fibonacci Structure and Euler Factorization Mehiri, El-Mehdi Combinatorics Discrete Mathematics Number Theory 05A17, 11P81, 05A15, 11P82, 11P84 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. |
| title | Block-Separated Overpartitions: Fibonacci Structure and Euler Factorization |
| topic | Combinatorics Discrete Mathematics Number Theory 05A17, 11P81, 05A15, 11P82, 11P84 |
| url | https://arxiv.org/abs/2511.16580 |