Abstract
Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let ℓ be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,ℓ) modulo any power of that prime ℓ α. In addition, we establish general congruence relations for p(n,ℓ) modulo ℓ α for any n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlgren, S.: Distribution of the partition function modulo composite integers M. Math. Ann. 318, 795–803 (2000)
Andrews, G.E.: The Theory of Partitions, the Encyclopedia of Mathematics and Its Applications Series. Addison-Wesley, New York (1976). Reissued, Cambridge University Press, New York (1998)
Atkin, A.O.L.: Congruence Heke Operators. Proc. Symp. Pure Math. 12, 33–40 (1969)
Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glascow Math. J. 8, 14–32 (1967)
Gupta, H.: Partitions—a survey. J. Res. Natl. Bureau Stand. Math. Sci. 74B(1) (1970)
Gupta, H., Gwyther, E.E., Miller, J.C.P.: Tables of Partitions. Royal Soc. Math. Tables, vol. 4. Cambridge University Press, Cambridge (1958)
Kronholm, B.: On Congruence Properties of p(n,m). PAMS 133, 2891–2895 (2005)
Kronholm, B.: On consecutive congruence properties of p(n,m). Integers 7, #A16 (2007)
Kwong, Y.H.: Minimum periods of binomial coefficients modulo M. Fibonacci Q., 27, 348–351 (1989)
Kwong, Y.H.: Minimum periods of partition functions modulo M. Util. Math. 35, 3–8 (1989)
Kwong, Y.H.: Periodicities of a class of infinite integer sequences modulo M. J. Number Theory 31, 64–79 (1989)
Nijenhuis, A., Wilf, H.S.: Periodicities of partition functions and Stirling numbers modulo p. J. Number Theory 25, 308–312 (1987)
Ono, K.: Distribution of the partition function modulo m. Ann. Math. 151, 293–307 (2000)
Ramanujan, S.: Congruence properties of partitions. Proc. Lond. Math. Soc. (2) 19, 207–210 (1919)
Ramanujan, S.: Collected Papers. Cambridge University Press, London (1927). Reprinted: AMS, Chelsea (2000) with new preface and extensive commentary by B. Berndt
Sylvester, J.J.: On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order. With an excursus on rational fractions and partitions. Am. J. Math. 5, 79–136 (1882)
Watson, G.N.: Ramanujans Vermutungüber Zerfällungsanzahlen. J. Reine Angew. Math. 179, 97–128 (1938)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kronholm, B. Generalized congruence properties of the restricted partition function p(n,m). Ramanujan J 30, 425–436 (2013). https://doi.org/10.1007/s11139-012-9382-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-012-9382-x