Consider Stirling numbers of the first kind $ s(i, j)$ , and form a matrix $ S_1(n),$ where the $ i, j$ th entry is $ s(i, j)$ (IMPORTANT NOTE the indices start at $ 0,$ so this matrix is $ (n+1)\times (n+1).$ ) This matrix is the inverse of the similar matrix for the Stirling numbers of the second kind, and the question applies to both of them.
Now, for the question: I have computed the logs of the singular values of these things (you have to do this to high precision, since the entries grow very fast). Here is the plot of the singular values for $ n=100:$
You will note two empirical facts: firstly, $ 1$ is a singular value, and secondly the singular values bigger than one decay exponentially, as do the ones smaller than one, but with a different rate. Has this been observed? Can one prove it (or at least explain it)?
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