# Proportion of partitions in a rectangle

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.