In the previous post we described a fairly straightforward argument, using generating functions and the saddle-point bound, for giving an upper bound
on the partition function . In this post I’d like to record an elementary argument, making no use of generating functions, giving a lower bound of the form for some , which might help explain intuitively why this exponential-of-a-square-root rate of growth makes sense.
The starting point is to think of a partition of as a Young diagram of size , or equivalently (in French coordinates) as a lattice path from somewhere on the y-axis to somewhere on the x-axis, which only steps down or to the right, such that the area under the path is . Heuristically, if the path takes a total of steps then there are about such paths, and if the area under the path is then the length of the path should be about , so this already goes a long way towards explaining the exponential-of-a-square-root behavior.