Congruences for \(\ell\)-regular partitions and bipartitions. (English) Zbl 1450.11108

The authors define the following \(q\)-series: \[ F(q)=\sum_{n=-\infty}^\infty(-1)^{\delta n}(an+b)q^{(cn^2+dn)/2}, \] where \(\delta, a, b, c, d\in\mathbb{Z}, c\neq0, c\equiv d\pmod{2}\), and \(a=2bc/d\neq0\) or \(a=0\). The function \(F(q)\) contains the Ramanujan general theta function \(f(a,b)=\sum\limits_{n=-\infty}^\infty a^{n(n-1)/2}b^{n(n+1)/2}\) \((|ab|<1)\) as a special case. Using standard \(q\)-series techniques, the authors derive an dissection identity for the coefficients of \(F(q)\) and use this identity to study congruence properties satisfied by the coefficients of \(F(q)\). As an immediate consequence, they obtain several infinite families of congruences for \(\ell\)-regular partition function and \(\ell\)-regular bipartition function. Moreover, based on the aforementioned dissection identity, the authors also give a new proof of the Ramanujan congruence for partition function modulo 5.


11P83 Partitions; congruences and congruential restrictions
05A17 Combinatorial aspects of partitions of integers
Full Text: DOI Euclid


[1] S. Ahlgren and J. Lovejoy, “The arithmetic of partitions into distinct parts”, Mathematika 48:1-2 (2001), 203-211. · Zbl 1040.11073 · doi:10.1112/S0025579300014443
[2] G. E. Andrews, M. D. Hirschhorn, and J. A. Sellers, “Arithmetic properties of partitions with even parts distinct”, Ramanujan J. 23:1-3 (2010), 169-181. · Zbl 1218.05018 · doi:10.1007/s11139-009-9158-0
[3] B. C. Berndt, Number theory in the spirit of Ramanujan, Student Mathematical Library 34, Amer. Math. Soc., Providence, RI, 2006. · Zbl 1117.11001
[4] S.-C. Chen, “On the number of partitions with distinct even parts”, Discrete Math. 311:12 (2011), 940-943. · Zbl 1232.11110 · doi:10.1016/j.disc.2011.02.025
[5] S.-P. Cui and N. S. S. Gu, “Arithmetic properties of \(\ell \)-regular partitions”, Adv. in Appl. Math. 51:4 (2013), 507-523. · Zbl 1281.05013 · doi:10.1016/j.aam.2013.06.002
[6] H. Dai, “Congruences for the number of partitions and bipartitions with distinct even parts”, Discrete Math. 338:3 (2015), 133-138. · Zbl 1303.11114 · doi:10.1016/j.disc.2014.10.013
[7] G. Gasper and M. Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications 96, Cambridge Univ. Press, 2004. · Zbl 1129.33005
[8] B. Gordon and K. Ono, “Divisibility of certain partition functions by powers of primes”, Ramanujan J. 1:1 (1997), 25-34. · Zbl 0907.11036 · doi:10.1023/A:1009711020492
[9] M. D. Hirschhorn and J. A. Sellers, “Elementary proofs of parity results for \(5\)-regular partitions”, Bull. Aust. Math. Soc. 81:1 (2010), 58-63. · Zbl 1248.11077 · doi:10.1017/S0004972709000525
[10] Q.-H. Hou, L. H. Sun, and L. Zhang, “Quadratic forms and congruences for \(\ell \)-regular partitions modulo \(3, 5\) and \(7\)”, Adv. in Appl. Math. 70 (2015), 32-44. · Zbl 1327.05025 · doi:10.1016/j.aam.2015.06.005
[11] B. L. S. Lin, “Arithmetic properties of bipartitions with even parts distinct”, Ramanujan J. 33:2 (2014), 269-279. · Zbl 1298.05033 · doi:10.1007/s11139-013-9473-3
[12] J. Lovejoy, “The number of partitions into distinct parts modulo powers of \(5\)”, Bull. London Math. Soc. 35:1 (2003), 41-46. · Zbl 1102.11056 · doi:10.1112/S0024609302001492
[13] K. Ono and D. Penniston, “The \(2\)-adic behavior of the number of partitions into distinct parts”, J. Combin. Theory Ser. A 92:2 (2000), 138-157. · Zbl 0972.05003 · doi:10.1006/jcta.2000.3057
[14] S. Ramanujan, “Some properties of \(p(n)\), the number of partitions of \(n\) [Proc. Cambridge Philos. Soc. 19 (1919), 207-210]”, pp. 210-213 in Collected papers of Srinivasa Ramanujan, edited by G. H. Hardy et al., Amer. Math. Soc., Providence, RI, 2000. Third printing of the 1927 original.
[15] O. · Zbl 1275.11136 · doi:10.1016/j.jnt.2012.11.002
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