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Let $n, j, k$ be three positive integers. Let $M(n,j,k)$ be the number of ways a set of $n$ elements can be partitioned into $j$ subsets, each of them containing at most $k$ elements.

What method should I use in order to understand the asymptotics of $M(n,j,k)$ when $n$ is large and $j$ and $k$ are some finite numbers?

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    Related: [Partitions in a rectangle](https://en.wikipedia.org/wiki/Partition_(number_theory)#Partitions_in_a_rectangle_and_Gaussian_binomial_coefficients) – 2017-02-22
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    I guess that *divided* stands for *partitioned*, am I correct? – 2017-02-22
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    for $k=n$, the answer is $\lbrace{n\atop j}\rbrace$, which is a "stirling number of second type". This is not the same as the binomial coefficient ${n}\choose{j}$ – 2017-02-22
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    @Joffan: they are not strictly related, since subsets with the same cardinality but different elements are not considered the same thing. This problem can be tackled through exponential generating functions and the saddle point method, see *Wilf - Generatingfunctionology* or *Flajolet - Analytic Combinatorics*. – 2017-02-22
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    Assuming @Simon is right (which is the only way I know how to make sense of this question) then I believe you want to use the mixed generating functions $f_k(x,y)=\exp\left(y\sum\limits_{i=1}^{k}\frac{1}{i!}x^i\right)$ for which $M(n,j,k)=[y^jx^n/n!]f_k(x,y)$. – 2017-02-22
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    I edited the question to avoid misunderstanding. I meant "partitioned". – 2017-02-22
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    @JackD'Aurizio yes, thanks for the additional thoughts and methods. – 2017-02-22

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