Congruences for simultaneously s-regular and t-regular partition function
DOI:
https://doi.org/10.5269/bspm.68794Abstract
A partition of a positive integer $n$ is said to be simultaneously $s$-regular and $t$-regular partition if none of the parts are divisible by $s$ and $t$. In this paper, we establish many infinite families of congruences for simultaneously $s$-regular and $t$-regular partition function by considering some particular values of $s$ and $t$.
References
1. B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
2. B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., 1995.
3. N. D. Baruah and K. K. Ojah, Partitions with designated summands in which all parts are odd, Integers, 15 (2015), #A9.
4. S.-P. Cui and N. S. S. Gu, Arithmetic properties of â„“-regular partitions, Adv. Appl. Math. 51 (2013), 507-523.
5. M. D. Hirschhorn and J. A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bull. Aust. Math. Soc. 81 (2010), 58-63.
6. M. D. Hirschhorn, The Power of q, A Personal Journey, Developments in Mathematics, Springer, Cham, 2017.
7. W. J. Keith, A bijection for partitions simultaneously s-regular and t-distinct, Integers, 23 (2023), #A9.
8. D. Penniston, Arithmetic of â„“-regular partition functions, Int. J. Number Theory 4 (2008), 295-302.
9. S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), 147-153.
10. S. Ramanujan and G. H. Hardy, Collected papers, Chelsea, New York, 1962.
11. J. J. Webb, Arithmetic of the 13-regular partition function modulo 3, Ramanujan J. 25 (2011), 49-56.
12. E. X. W. Xia and O. X. M. Yao, Some modular relations for the G¨ ollnitz-G¨ ordon functions by an even-odd method, J. Math. Anal. Appl. 387 (2012), 126-138.
13. E. X. W. Xia and O. X. M. Yao, Analogues of Ramanujan’s partition identities, Ramanujan J. 31 (2013), 373-396.
2. B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., 1995.
3. N. D. Baruah and K. K. Ojah, Partitions with designated summands in which all parts are odd, Integers, 15 (2015), #A9.
4. S.-P. Cui and N. S. S. Gu, Arithmetic properties of â„“-regular partitions, Adv. Appl. Math. 51 (2013), 507-523.
5. M. D. Hirschhorn and J. A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bull. Aust. Math. Soc. 81 (2010), 58-63.
6. M. D. Hirschhorn, The Power of q, A Personal Journey, Developments in Mathematics, Springer, Cham, 2017.
7. W. J. Keith, A bijection for partitions simultaneously s-regular and t-distinct, Integers, 23 (2023), #A9.
8. D. Penniston, Arithmetic of â„“-regular partition functions, Int. J. Number Theory 4 (2008), 295-302.
9. S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), 147-153.
10. S. Ramanujan and G. H. Hardy, Collected papers, Chelsea, New York, 1962.
11. J. J. Webb, Arithmetic of the 13-regular partition function modulo 3, Ramanujan J. 25 (2011), 49-56.
12. E. X. W. Xia and O. X. M. Yao, Some modular relations for the G¨ ollnitz-G¨ ordon functions by an even-odd method, J. Math. Anal. Appl. 387 (2012), 126-138.
13. E. X. W. Xia and O. X. M. Yao, Analogues of Ramanujan’s partition identities, Ramanujan J. 31 (2013), 373-396.
Downloads
Published
2025-05-28
Issue
Section
Research Articles
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
