Here is another proof and perspective for this result:

Partitions are in bijection with monomial ideals, $I$ in $S=\mathbb k[x,y]$ where $S/I$ is finite dimensional. Suppose we wanted to come up with a model that selected a random monomial ideal. One of the most natural things to try is an Erdos-Renyi type model: We can go over every monomial and choose to include it in the generating set for $I$ with probability $p$.

If you do this, then the probability that a random ideal is the monomial ideal corresponding to partition $\lambda$ is given by $p^{|\lambda|}(1-p)^{m(\lambda)}$, where $m(\lambda)$ denotes the number of minimal generators for the ideal corresponding to $\lambda$. This is the same as the number of corners as you define them.

You can learn more about this sort of thing in "Random monomial ideals" by J.A. De Loera, S. Petrović, L. Silverstein, D. Stasi, D. Wilburne.

Here is another proof and perspective for this result:

Partitions are in bijection with monomial ideals, $I$ in $S=\mathbb k[x,y]$ where $S/I$ is finite dimensional. Suppose we wanted to come up with a model that selected a random monomial ideal. One of the most natural things to try is an Erdos-Renyi type model: We can go over every monomial and choose to exclude it from the generating set for $I$ with probability $p$.

If you do this, then the probability that a random ideal is the monomial ideal corresponding to partition $\lambda$ is given by $p^{|\lambda|}(1-p)^{m(\lambda)}$, where $m(\lambda)$ denotes the number of minimal generators for the ideal corresponding to $\lambda$. This is the same as the number of corners as you define them.

You can learn more about this sort of thing in "Random monomial ideals" by J.A. De Loera, S. Petrović, L. Silverstein, D. Stasi, D. Wilburne.