Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910051433709568 |
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| author | Hussain, Imdadul Ghoshal, Suparno Jana, Arijit |
| author_facet | Hussain, Imdadul Ghoshal, Suparno Jana, Arijit |
| contents | In a recent paper, Thejitha and Fathima introduced the overcolored partition function $\bar{a}_{r,s}(n)$, which enumerates overpartitions in which even parts may appear in one of $r$ colors and odd parts in one of $s$ colors, for fixed integers $r,s \geq 1$. They also proposed several conjectures concerning families of congruences modulo powers of $2$ for specific arithmetic progressions of $\bar{a}_{r,s}(n)$. In this paper, we provide an elementary proof of this conjecture that relies only on classical $q$-series manipulations and properties of Ramanujan's theta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12401 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts Hussain, Imdadul Ghoshal, Suparno Jana, Arijit Number Theory In a recent paper, Thejitha and Fathima introduced the overcolored partition function $\bar{a}_{r,s}(n)$, which enumerates overpartitions in which even parts may appear in one of $r$ colors and odd parts in one of $s$ colors, for fixed integers $r,s \geq 1$. They also proposed several conjectures concerning families of congruences modulo powers of $2$ for specific arithmetic progressions of $\bar{a}_{r,s}(n)$. In this paper, we provide an elementary proof of this conjecture that relies only on classical $q$-series manipulations and properties of Ramanujan's theta function. |
| title | Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.12401 |