Stirling Numbers
Vector 9:1, page 129
Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 129
I am delighted that Norman Thomson and Ken Iverson were interested enough to comment on my article on Stirling numbers. As a result, I have learned enough J to understand and execute the wonderfully concise and interesting expression in the latter’s letter, thanks to my colleague Chris Brunsdon. Unfortunately, I ran out of workspace on my micro long before I could execute it with a sufficiently high parameter value (60) to compare the results with those of the programs I presented.
It might help anyone interested in this correspondence if I gave an APL version of the method (index origin is zero as in J).
The relationship between the two kinds of Stirling number is certainly made very clear, though as a practical method of calculation it is not as useful as Norman’s recursive function.
Perhaps I should come clean! I wrote the original article because the expression ⍎1↓(7×P)⍴'×+\(⍳N)' amused me, and I thought it might amuse others. I am sure that J can produce the same sort of expression without using character vectors, but I am afraid I cannot yet construct it. However, I also had a real problem involving Stirling numbers, or, as it turned out, the ratios of Stirling numbers, and I needed a good algorithm. Recursion seems to fit the bill best, and my eventual function to calculate \(t\sigma^{n}_{t-1}/\sigma^{n}_{t}\) was (with index origin back to 1):
If this looks very like Norman’s function, that is because I modelled the final version on it!