I imagine someone has studied this already so I'm just looking for resources: what sort of asymptotic behavior does the distribution of integer partitions of $n$ counted by the number of parts exhibit? Looking at the plots that WolframAlpha supplies for low-ish values (around 30), the distribution looks sort of like Tracy-Widom.
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Does this answer your question: https://math.stackexchange.com/a/3670073/123905 ? – Eric Towers Oct 29 '21 at 06:19
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@EricTowers yes, thank you – failedentertainment Oct 29 '21 at 06:30