Parameterization of Irreducible Representations of $\text{GL}_m(\mathbb{C})$

Question

Here is an excerpt from the beginning of Chapter 8 of Fulton's Young Tableaux: .

I do not understand how we can parameterize just be diagrams $\lambda$ with at most $m$ rows? What about the length of the rows? I feel like this should read "diagrams $\lambda \vdash n$ with at most $m$ rows" but then I do not know what $n$ should be.