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 $\{1,\dots,n\}$ 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 $\{1,\dots,n\}$ is $2\sqrt{n}$. For more along these lines, see Richard Stanley's 2006 ICM talk.

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 Increasing and decreasing subsequences and their variants.

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 $\{1,\dots,n\}$ 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 $\{1,\dots,n\}$ is $2\sqrt{n}$. For more along these lines, see Richard Stanley's 2006 ICM talk.

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 $\{1,\dots,n\}$ 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 $\{1,\dots,n\}$ is $2\sqrt{n}$. For more along these lines, see Richard Stanley's 2006 ICM talk.