Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908773937840128 |
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| author | Johnson, Anjelin Mariya Sellers, James A. Fathima, S. N. |
| author_facet | Johnson, Anjelin Mariya Sellers, James A. Fathima, S. N. |
| contents | Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers Johnson, Anjelin Mariya Sellers, James A. Fathima, S. N. Number Theory 05A17, 05A15, 11P83 Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren. |
| title | Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers |
| topic | Number Theory 05A17, 05A15, 11P83 |
| url | https://arxiv.org/abs/2510.25250 |