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
>
