Let $ k, n, r \geqslant 1 $ be integers. Let $ \lambda $ be a partition of $r$, what we denote by $ \lambda \vdash r $.

I would like a lower and an upper bound for the following quantity, for all $ r \leqslant nk $

$$ b_r(n, k) := \frac{ \# \left\{ \lambda \vdash r : \lambda \subset (n^k) \right\} }{ \# \left\{ \lambda \vdash r \right\} } $$

Here, $ \lambda \subset (n^k) $ means that $ \ell(\lambda) \leqslant k $ and $ \lambda_1 \leqslant n $, namely, that $ \lambda $ is included in the rectangular partition $ (n^k) $.

For instance, for $ r = nk $, we have $ \# \left\{ \lambda \vdash nk : \lambda \subset (n^k) \right\} = 1 $ (only the rectangle has the right size), so the value is $ 1 / p(nk) $ with with $ p(r) := \# \left\{ \lambda \vdash r \right\} $.

This can be seen as the probability $ \mathbb{P}\!\left( \boldsymbol{\lambda} \subset (n^k) \right) $ where $ \boldsymbol{\lambda} $ is uniformly distributed amongst the partitions of size $r$ (i.e. all partitions have equal probability $ 1/p(r) $ to be selected). This measure has been investigated by Vershik and Yakubovich amongst others.

So far, I had a look at Andrew's book "the theory of partitions". There is a whole chapter on asymptotics of infinite product generating functions (the chapter 6) with a meta theorem due to Meinardus. Unfortunately, the generating series of partitions included in a rectangle is given by a Gaussian polynomial, hence a ratio and not only a product, and Meinardus' theorem does not apply. Nevertheless, I am not updated on the recent research on the topic, so a more recent theorem may solve the problem.