Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. Here is yet another cute finding for which I ask a:

Question. Is there a direct proof (or interpretation or transformation) of the following equality?

$$\det\left[I_{i+j}\right]_{i,j=0}^n =\det\left[(i+1)^j\right]_{i,j=0}^n.$$

Remark 1. I know how to prove them individually, so that's not what we seek here. The RHS is a specialization of Vandermonde.

Remark 2. One may consider looking at the two MO questions/answers: here and also here.


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