Fix an integer $n\geq 8$. For each integer $i\leq n/2$, denote by $X_i$ the set $$X_i = \left\{ \frac{n-i+1-k}{n-i+1}\binom nk\binom{k-1}{i-1} ~\middle|~~ i\leq k\leq n-i\right\}.$$

**The question:**

Are the sets $X_1$ and $X_2$ disjoint?

**The reason:**

The product $\frac{n-i+1-k}{n-i+1}\binom nk\binom{k-1}{i-1}$ is the degree in the symmetric group $S_n$ of the irreducible character which corresponds to the partition $(n-k,i,1^{k-i})$ (the latter entry meaning the cell $1$ repeated $k-i$ times), and a conjecture on which I am working will substantively advance if I get the full range of values coming from $X_1\cup X_2$ without having to worry about whether or not I have repetition by pulling something from $X_1$ then something from $X_2$.

**The bonus question:**

What I really want to be true is that $X_1$, $X_2$, and $X_3$ are pairwise disjoint, so a proof that covers all 3 pairs simultaneously would be the gold standard, but I would not at all be surprised if each $X_i,X_j$ pair had something peculiar to it that required individualized effort. For instance, one reason $n\geq 8$ is required is that the sets $X_2$ and $X_3$ have respectively corresponding partitions $(5,2)$ and $(4,3)$ of $n=7$ which both yield the integer $14$; they are not disjoint.

**Potentially relevant information:**

I readily note that, in this indexing, both $X_1$ and $X_2$ have internal redundancy, i.e., if viewed as multisets, there would be multiple appearances of some integers. I'm only asking about *between* sets, not within a given set.

**My work:**

Computer runs up into the $n=3200$ range say they are disjoint at least that far.