Abstract:   Let p_r(n) denote the number of r-component multipartitions of n, and let S_{γ λ} be the space spanned by η(24z)^γφ(24z), where η(z) is the Dedekind's eta function and φ(z) is a holomorphic modular form in M_λ(SL₂(Z)). In this paper, we show that the generating function of p_r with respect to n is congruent to a function in the space S_{γ, λ} modulo m^k. As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5, 7, 11 and Gandhi's congruences of p₂(n) modulo 5 and p₈(n) modulo 11. Furthermore, using the invariance property of S_{γ, λ} under the Hecke operator , we obtain two classes of congruences pertaining to the m^k-adic property of p_r(n).

 AMS Classification:  05A17, 11F33, 11P83.

 Keywords:   modular form, partition, multipartition, Ramanujan-type congruence

 Download:   pdf