On 3- and 9-regular overpartitions modulo powers of 3. (English) Zbl 1423.11180
Summary: Let \(\overline { A}_3(n)\) and \(\overline { A}_9(n)\) denote the number of 3- and 9-regular overpartitions of \(n\). For each \(\alpha > 0\), we obtain the generating functions for \(\overline { A}_{3}(3^{2\alpha}n)\), \(\overline { A}_{3}(3^{2\alpha -1}n)\) and \(\overline { A}_{9} (3^{\alpha}n)\). We show that \(\overline { A}_{3}(n)\) and \(\overline { A}_{9}(n)\) satisfy certain internal congruences. (10 Refs.)
MSC:
| 11P83 | Partitions; congruences and congruential restrictions |
| 05A17 | Combinatorial aspects of partitions of integers |
References:
| [1] | A. M. Alanazi, A. O. Munagi, and J. A. Sellers, An infinite family of congruences for ‘-regular overpartitions, Integers 16 (2016), #A37. [A]G. E. Andrews, Singular overpartitions, Int. J. Number Theory 11 (2015), 1523- 1534. [B]B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer, 1991. [CHS] S. Chen, M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of Andrews’ singular overpartitions, Int. J. Number Theory 11 (2015), 1463-1476. [CL]S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623-1635. |
| [2] | M. D. Hirschhorn and D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math. 326 (1981), 1-17. · Zbl 0452.10015 |
| [3] | M. D. Hirschhorn and J. A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput. 53 (2005), 65-73. · Zbl 1086.11048 |
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