Let $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $ denote the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct odd parts and the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct even parts, respectively. Our first goal in this note was to prove two congruence relations for $ {\rm{pod}}_\ell(n) $. Furthermore, we found a formula for the action of the Hecke operator on a class of eta-quotients. As two applications of this result, we obtained two infinite families of congruence relations for $ {\rm pod}_5(n) $. We also proved a congruence relation for $ {\rm{ped}}_\ell(n) $. In particular, we established a congruence relation modulo 2 connecting $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $.
Citation: JingJun Yu. Some congruences for $ \ell $-regular partitions with certain restrictions[J]. AIMS Mathematics, 2024, 9(3): 6368-6378. doi: 10.3934/math.2024310
Let $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $ denote the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct odd parts and the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct even parts, respectively. Our first goal in this note was to prove two congruence relations for $ {\rm{pod}}_\ell(n) $. Furthermore, we found a formula for the action of the Hecke operator on a class of eta-quotients. As two applications of this result, we obtained two infinite families of congruence relations for $ {\rm pod}_5(n) $. We also proved a congruence relation for $ {\rm{ped}}_\ell(n) $. In particular, we established a congruence relation modulo 2 connecting $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $.
[1] | M. D. Hirschhorn, J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, J. Ramanujan J., 22 (2010), 273–284. http://doi.org/10.1007/s11139-010-9225-6 doi: 10.1007/s11139-010-9225-6 |
[2] | S. Radu, J. A. Sellers, Congruence properties modulo 5 and 7 for the pod function, Int. J. Number Theory, 07 (2011), 2249–2259. http://doi.org/10.1142/S1793042111005064 doi: 10.1142/S1793042111005064 |
[3] | J. Lovejoy, R. Osburn, Quadratic forms and four partition functions modulo 3, Integers 11, 04 (2011), 47–53. https://doi.org/10.1515/integ.2011.004 doi: 10.1515/integ.2011.004 |
[4] | S. P. Cui, W. X. Gu, Z. S. Ma, Congruences for partitions with odd parts distinct modulo 5, Int. J. Number Theory, 11 (2015), 2151–2159. https://doi.org/10.1142/S1793042115500943 doi: 10.1142/S1793042115500943 |
[5] | H. G. Fang, F. G. Xue, X. M. Yao, New congruences modulo 5 and 9 for partitions with odd parts distinct, Quaest. Math., 43 (2020), 1573–1586. https://doi.org/10.2989/16073606.2019.1653394 doi: 10.2989/16073606.2019.1653394 |
[6] | D. S. Gireesh, M. D. Hirschhorn, M. S. Mahadeva Naika, On 3-regular partitions with odd parts distinct, Ramanujan J., 44 (2017), 227–236. https://doi.org/10.1007/s11139-016-9814-0 doi: 10.1007/s11139-016-9814-0 |
[7] | N. Saika, Infinite families of congruences for 3-regular partitions with distinct odd parts, Commun. Math. Stat., 8 (2020), 443–451. https://doi.org/10.1007/s40304-019-00182-7 doi: 10.1007/s40304-019-00182-7 |
[8] | V. S. Veena, S. N. Fathima, Arithmetic properties of 3-regular partitions with distinct odd parts, Abh. Math. Semin. Univ. Hambg., 91 (2021), 69–80. https://doi.org/10.1007/s12188-021-00230-6 doi: 10.1007/s12188-021-00230-6 |
[9] | R. Drema, N. Saikia, Arithmetic properties for $l$-regular partition functions with distinct even parts, Bol. Soc. Mat. Mex., 28 (2022), 10–20. https://doi.org/10.1007/s11139-018-0044-5 doi: 10.1007/s11139-018-0044-5 |
[10] | B. Hemanthkumar, H. S. Sumanth Bharadwaj, M. S. Mahadeva Naika, Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts, Acta Mathematica Vietnamica, 44 (2019), 797–811. https://doi.org/10.1007/s40306-018-0274-z doi: 10.1007/s40306-018-0274-z |
[11] | J. Tate, The non-existence of certain Galois extensions of $\mathbb{Q}$ unramified outside 2, Contemp. Math., 174 (1994), 153–156. https://doi.org/10.1090/conm/174/01857 doi: 10.1090/conm/174/01857 |
[12] | K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-Series, Providence: American Mathematical Society, 2004. http://dx.doi.org/10.1090/cbms/102 |
[13] | K. Mahlburg, More congruences for the coefficients of quotients of Eisentein series, J. Number Theory, 115 (2005), 89–99. https://doi.org/10.1016/j.jnt.2004.10.008 doi: 10.1016/j.jnt.2004.10.008 |
[14] | M. Boylan, Congruences for $2^t$-core partition functions, J. Number Theory, 92 (2002), 131–138. https://doi.org/10.1006/jnth.2001.2695 doi: 10.1006/jnth.2001.2695 |
[15] | M. D. Hirschhorn, J. A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bull. Aust. Math. Soc., 81 (2010), 58–63. https://doi.org/10.1017/S0004972709000525 doi: 10.1017/S0004972709000525 |