Congruences modulo powers of 5 for the crank parity function

Abstract

In 1988, Andrews and Garvan introduced the partition statistic “crank” in order to give combinatorial interpretation for Ramanujan’s  celebrated partition congruence modulo 11. In 2009, Choi, Kang and Lovejoy established congruences modulo powers of 5 for the crank  parity function pC (n) by utilizing the theory of modular forms, where pC (n) denotes the difference between the number of partitions of  n with even crank and the number of partitions of n with odd crank. In this paper, we provide a completely elementary proof of this  congruence family. Moreover, we also prove several new congruences modulo small powers of 5.