Proof of a Conjecture of Banerjee and Dastidar on Odd Crank

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 cₒ(n) , the number of partitions of n with odd crank. They conjectured a congruence modulo $4$ satisfied by cₒ(n) . We confirm the conjecture and evaluate cₒ(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.