On \((\ell , m)\)-regular partitions with distinct parts
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Abstract
Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example, where \(\alpha , \beta \ge 0\).
$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$
Keywords
Partition identities Theta-functions Partition congruences Regular partitionMathematics Subject Classification
11P83 05A17Notes
Acknowledgements
The authors are thankful to the referee for useful suggestions which improved the quality of the paper.
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