Open problems involving cranks for partition congruences

Abstract: Dyson famously conjectured, correctly, that his rank statistic witnesses Ramanujan's first two congruences for $p(n)$, and that there exists a ``crank" statistic that witnesses Ramanujan's congruence modulo $11$ in a similar fashion. As it turns out, this phenomenon of congruence-witnessing statistics, which we now also call ``cranks" in homage to Dyson, also occurs in other contexts within partition theory. In this talk, we give several open problems and conjectures in this area, highlighting some recent developments along the way.\\ This talk will include joint work with several coauthors.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)