A proposed crank for $(k+j)$-colored partitions, with $j$ colors having distinct parts
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910521726337024 |
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| author | Wilson, Samuel |
| author_facet | Wilson, Samuel |
| contents | In 1988, George Andrews and Frank Garvan discovered a crank for $p(n)$. In 2020, Larry Rolen, Zack Tripp, and Ian Wagner generalized the crank for p(n) in order to accommodate Ramanujan-like congruences for $k$-colored partitions. In this paper, we utilize the techniques used by Rolen, Tripp, and Wagner for crank generating functions in order to define a crank generating function for $(k + j)$-colored partitions where $j$ colors have distinct parts. We provide three infinite families of crank generating functions and conjecture a general crank generating function for such partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A proposed crank for $(k+j)$-colored partitions, with $j$ colors having distinct parts Wilson, Samuel Combinatorics Number Theory 05A15 (Primary), 05A17 (Secondary) In 1988, George Andrews and Frank Garvan discovered a crank for $p(n)$. In 2020, Larry Rolen, Zack Tripp, and Ian Wagner generalized the crank for p(n) in order to accommodate Ramanujan-like congruences for $k$-colored partitions. In this paper, we utilize the techniques used by Rolen, Tripp, and Wagner for crank generating functions in order to define a crank generating function for $(k + j)$-colored partitions where $j$ colors have distinct parts. We provide three infinite families of crank generating functions and conjecture a general crank generating function for such partitions. |
| title | A proposed crank for $(k+j)$-colored partitions, with $j$ colors having distinct parts |
| topic | Combinatorics Number Theory 05A15 (Primary), 05A17 (Secondary) |
| url | https://arxiv.org/abs/2407.07891 |