Congruences for 13-regular partitions

Abstract

A partition is called ℓ -regular partition if there is no part divisible by ℓ. Let bℓ (n) denote the number of ℓ -regular partitions of n. The divisibility and distribution of b ℓ (n) have been widely studied in the literature. In this paper, we mainly focus on congruences for the 13-regular partition function. By means of Jacobi's identity and the modular equations of fifth and seventh order, we derive new infinite families of congruences for b₁₃(n) modulo 13. Meanwhile, in view of a congruence relation for b₁₃(n) given by Calkin et al. and some congruences for b4(n), we obtain further congruences for b₁₃(n) modulo 2. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.