Proof of a Conjecture of Banerjee and Dastidar on Odd Crank

Abstract

Recently, when studying intricate connections between Ramanujan’s theta functions and a class of partition functions, Banerjee and Dastidar [‘Ramanujan’s theta functions and parity of parts and cranks of partitions’, Ann. Comb., to appear] studied some arithmetic properties for $c_o(n)$ , the number of partitions of n with odd crank. They conjectured a congruence modulo $4$ satisfied by $c_o(n)$ . We confirm the conjecture and evaluate $c_o(4n)$ modulo $8$ by dissecting some q-series into even powers. Moreover, we give a conjecture on the density of divisibility of odd cranks modulo 4, 8 and 16.