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PROOF OF A CONJECTURE OF BANERJEE AND DASTIDAR ON ODD CRANK
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 3 / December 2023
- Published online by Cambridge University Press:
- 13 January 2023, pp. 406-411
- Print publication:
- December 2023
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- Article
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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.
