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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.04402 |
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| _version_ | 1866909677144506368 |
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| author | Donato, Judy Ann |
| author_facet | Donato, Judy Ann |
| contents | A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer that is not a part of the partition. In 2024, Aricheta and Donato studied the minimal excludant of the non-overlined parts of an overpartition, where an overpartition of $n$ is a partition of $n$ in which the first occurrence of a number may be overlined. In this research, we explore two other definitions of the minimal excludant of an overpartition: (i) considering only the overlined parts, and (ii) considering both the overlined and non-overlined parts. We discuss the combinatorial, asymptotic, and arithmetic properties of the corresponding $σ$-function, which gives the sum of the minimal excludants over all overpartitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Minimal Excludant over Overpartitions Donato, Judy Ann Number Theory 05A17, 11F11, 11F20, 11P83 A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer that is not a part of the partition. In 2024, Aricheta and Donato studied the minimal excludant of the non-overlined parts of an overpartition, where an overpartition of $n$ is a partition of $n$ in which the first occurrence of a number may be overlined. In this research, we explore two other definitions of the minimal excludant of an overpartition: (i) considering only the overlined parts, and (ii) considering both the overlined and non-overlined parts. We discuss the combinatorial, asymptotic, and arithmetic properties of the corresponding $σ$-function, which gives the sum of the minimal excludants over all overpartitions. |
| title | On Minimal Excludant over Overpartitions |
| topic | Number Theory 05A17, 11F11, 11F20, 11P83 |
| url | https://arxiv.org/abs/2507.04402 |