Ramanujan J (2009) 19: 63–70 DOI 10.1007/s11139-007-9042-8 ℓ-Divisibility of ℓ-regular partition functions Brian Dandurand · David Penniston Received: 5 June 2007 / Accepted: 18 June 2007 / Published online: 19 October 2007 © Springer Science+Business Media, LLC 2007 Abstract We give exact criteria for the ℓ-divisibility of the ℓ-regular partition func- tion b ℓ (n) for ℓ ∈{5, 7, 11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a con- dition given by J.-P. Serre for the vanishing of the coefficients of  ∞ m=1 (1 − q m ) ℓ−1 . Keywords Partitions · Hecke eigenforms Mathematics Subject Classification (2000) 11P83 1 Introduction A partition of n is a non-increasing sequence of positive integers whose sum is n. For ℓ> 1 we call a partition ℓ-regular provided it has no summand divisible by ℓ. We will as usual denote by p(n) the number of partitions of n, and by b ℓ (n) the number of ℓ-regular partitions of n. Elementary techniques in the theory of partitions give the generating functions ∞  n=0 p(n)q n = ∞  m=1  1 1 − q m  B. Dandurand Department of Mathematics, Clemson University, Clemson, SC 29634, USA e-mail: bdandur@clemson.edu D. Penniston ( ) Department of Mathematics, Furman University, Greenville, SC 29613, USA e-mail: david.penniston@furman.edu