On 2019-02-11 11:07, Veit Elser wrote:

If you uniformly sample from the partitions of an integer N, and interpret the sequence of parts, sorted largest to smallest, as a (decreasing) function y(x), then in the limit of large N the y(x) of the “typical" partition satisfies (after rescaling)

(e^x-1)(e^y-1) = 1

Providing a somewhat unlikely looking involution: Out[755]= Function[x, x - Log[-1 + E^x]] In[760]:= NestList[%755, a, 9] // FullSimplify; In[761]:= Assuming[a > 0, FullSimplify@%] Out[761]= {a, a - Log[-1 + E^a], a, a - Log[-1 + E^a], a, a - Log[-1 + E^a], a, a - Log[-1 + E^a], a, a - Log[-1 + E^a]} (Also works for a:=±i, i+1, etc., but not -1.)

Does this curve have a name?

Hyperbolic hyperbola? HyperPERbola for short. Perhybola?

-Veit