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Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

Abstract : Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions of $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families of congruences modulo arbitrary powers of $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $n$ and $k$, we have \begin{align*} b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\mathrm{mod}~3^{2k}\right),\\ b_{27;3}\left(3^{2k+3}\cdot n+\dfrac{3^{2k+4}-13}{4}\right)&\equiv0~\left(\mathrm{mod}~3^{2k+5}\right). \end{align*}
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Submitted on : Monday, December 20, 2021 - 10:11:28 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM

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Nayandeep Deka Baruah, Hirakjyoti Das. Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2022, Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, pp.101 -- 115. ⟨10.46298/hrj.2022.8927⟩. ⟨hal-03498213⟩

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