There is a beautiful interpretation of $f(\chi)$ (that is to say, of the length of the first column of the partition), though it isn't very representation-theoretic.  

One way to generate Plancherel measure on partitions is to take uniformly at random a permutation of $\{1,\dotsc,n\}$ and apply Robinson–Schensted–Knuth to get a pair of Young tableaux of the same shape, and then take the partition encoded by that shape.  

The length of the first column in the shape corresponding to a permutation $\pi$ is the length of the longest decreasing sequence in $\pi$, while the length of the first row is the length of the longest increasing sequence in $\pi$.  

So Kerov–Vershik says (among other things) that the length of the longest decreasing sequence in a random permutation of $\{1,\dotsc,n\}$ is $2\sqrt{n}$.  For more along these lines, see 
Richard Stanley's 2006 ICM talk <a href="https://math.mit.edu/~rstan/papers/ids.pdf">Increasing and decreasing subsequences and their variants</a>.