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As promised, I've upgraded my last question. Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...

Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it. Consider the one-line partition $\lambda_n=(n)$ and its ...