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New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands

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Abstract

Recently, Lin introduced two new partition functions \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), which count the total number of tagged parts over all partitions of n with designated summands and the total number of tagged parts over all partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), and conjectured some congruences modulo 8. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2.

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Acknowledgements

The authors would like to thank the referee for his/her helpful comments and suggestions which helped improving the presentation of the paper.

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  1. Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam, 784028, India

    Nayandeep Deka Baruah & Mandeep Kaur

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  1. Nayandeep Deka Baruah
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  2. Mandeep Kaur
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Correspondence to Nayandeep Deka Baruah.

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Baruah, N.D., Kaur, M. New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands. Ramanujan J 52, 253–274 (2020). https://doi.org/10.1007/s11139-018-0112-x

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  • DOI: https://doi.org/10.1007/s11139-018-0112-x

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