Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts

Abstract

Let \(\text {pod}_{9}(n)\), \(\text {ped}_{9}(n)\), and \(\overline {A}_{9}(n)\) denote the number of 9-regular partitions of n wherein odd parts are distinct, even parts are distinct, and the number of 9-regular overpartitions of n, respectively. By considering \(\text {pod}_{9}(n)\) from an arithmetic point of view, we establish a number of infinite families of congruences modulo 16 and 32, and some internal congruences modulo small powers of 3. A relation connecting above partition functions in arithmetic progressions is obtained as follows. For any \(n\geq 0\),

$ 6 \text {pod}_{9}(2n + 1) = 2 \text {ped}_{9}(2n + 3) = 3 \overline {A}_{9}(n + 1).$

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Acknowledgements

The authors would like to thank anonymous referee for his/her valuable comments.

Funding

The second author was supported by the Council of Scientific and Industrial Research, India through SRF (No. 09/039(0111)/2014-EMR-I).

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Correspondence to M. S. Mahadeva Naika.

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Hemanthkumar, B., Bharadwaj, H.S.S. & Naika, M.S.M. Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts. Acta Math Vietnam 44, 797–811 (2019). https://doi.org/10.1007/s40306-018-0274-z

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Keywords

  • Regular partitions
  • Overpartitions
  • Congruences
  • Theta function
  • Distinct odd parts

Mathematics Subject Classification (2010)

  • 11P83
  • 05A17