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As the title suggests, I would like to compute probability vectors with elements proportional to (unsigned) Stirling numbers of the first kind by row. For easy reference, here is the Wiki page.

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For example, I'd like an efficient algorithm to return $p_6 \approx \frac{[120, \;274, \;225, \;85, \;15, \;1]}{720}$ when $n=6$. Previous MO questions have considered computing Sterling numbers, for example here and here. In my case, however, a reasonable approximation would suffice. I suppose that the scaling by row-sums might actually make the problem easier, as many of the elements decay rapidly with the row number $n$. It is not immediately obvious how to find such an approximation however, so I'm hoping someone here has a reference or some insight to share.

The approach of this paper might prove useful, but I have not digested it as yet.


This came up for me in a the course of a project concerning de Finetti theorems and kernel density estimation procedures, and is well outside my wheelhouse. I would be happy to hear that this question is trivial for combinatorics specialists. I have also tried the usual Google tactics, but precisely because there is a wealth of information about Stirling numbers I have found my question hard to address.

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I was able to find a reasonably good approximation using bounds given in a tech report of Jim Pitman.

Starting on page 13 an example is worked out which computes bounds on ratios of row-adjacent Stirling numbers using stable and efficient polygamma evaluations; see expressions (33) and (34) specifically.

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