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On congruence properties of the restricted partition functions \(p_{\omega }(n,k)\) and \(p_{\nu }(n,k)\)

  • Robson da Silva
  • Kelvin Souza de Oliveira
  • Almir Cunha da Graça Neto
Article

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.

Keywords

Partition Identities Ramanujan-type congruence 

Mathematics Subject Classification

11P81 11P82 11P83 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universidade Federal de São PauloSão José dos CamposBrazil
  2. 2.Universidade Federal do AmazonasManausBrazil
  3. 3.Universidade do Estado do AmazonasManausBrazil

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