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. 



Proportion of partitions in a rectangle
For many purposes, you might be satisfied with just the distribution of the largest part, which was studied by Erdős and Lehner, "The distribution of the number of summands in the partitions of a positive integer." Duke Math. J. 8 (1951) 335. Asymptotically, it has a Gumbel distribution. 


For the joint distribution of the largest part and number of parts, see G. Szekeres, "Asymptotic Distribution of partitions by number and size of parts." Colloquia Mathematica Societatis, János Bolyai, Number Theory, Budapest, 51 (1987), 527-538. I can't find this online but I can find the related results for partitions into distinct parts, [G. Szekeres, "Asymptotic Distribution of the Number and Size of Parts in Unequal Partitions." Bull Aust. Math. Soc. (1987) 89-97](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700026320). In a [paper](https://www.deepdyve.com/lp/springer-journals/limit-distributions-of-smallest-gap-and-largest-repeated-part-in-SQ93QoGSpd), [S. Wagner](http://math.sun.ac.za/~swagner/limitdist.pdf) refers to the results of Szekeres as follows: "Szekeres [17, 18] refined the result of Erdős and Lehner and also studied the joint distribution of length and maximum [20] (it turns out that, around their mean, the two are essentially independent of each other)."


