Abstract
In a recent work by Andrews, Dixit, and Yee the partition functions \(p_{\omega }(n)\) and \(p_{\nu }(n)\) were introduced in connection with the third order mock theta functions \(\omega (q)\) and \(\nu (q)\), respectively. The function \(p_{\omega }(n)\) counts the number of partitions of n in which each odd part is less than twice the smallest part, and \(p_{\nu }(n)\) counts the number of partitions of n under the same conditions as \(p_{\omega }(n)\) and having all parts distinct. In this paper, we consider restrictions of these functions, namely \(p_{\omega }(n,k)\) and \(p_{\nu }(n,k)\), where k is the number of parts. We present congruence properties for these restricted partition functions and we obtain classes of Ramanujan-type congruences for them.
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Acknowledgements
The authors would like to thank George Andrews for his helpful suggestions and comments. The authors also thank the anonymous referee for his/her helpful comments and suggestions.
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Robson da Silva was supported by FAPESP. Kelvin Souza de Oliveira and Almir Cunha da Graça Neto were supported by FAPEAM.
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da Silva, R., de Oliveira, K.S. & da Graça Neto, A.C. On congruence properties of the restricted partition functions \(p_{\omega }(n,k)\) and \(p_{\nu }(n,k)\). Ramanujan J 49, 105–113 (2019). https://doi.org/10.1007/s11139-018-0014-y
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DOI: https://doi.org/10.1007/s11139-018-0014-y