Bell numbers in Vector 17.1

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 112

In Joseph De Kerf’s “Bell Numbers and APL” in Vector 17.1 he writes, “Bell numbers are rather unknown….In most books on number theory…they are not mentioned, and this notwithstanding their practical importance.” It must be something of a surprise, then, to find two articles that discuss Bell numbers in the same issue of your nonmathematical magazine: Mr. De Kerf’s and mine (“Blists in OLEIS”). They are the central topic of his article, but only appear tangentially in mine. I drew on Kenneth Iverson’s discussion of the Stirling subset numbers in chapter 6, page 89, of his “Concrete Math Companion”. His lovely derivation uses matrix divide with powers on the left and falling factorials on the right to produce the table of Stirling subset numbers, whose row sums in turn give the Bell numbers. In his appendix Mr. De Kerf discusses this use of summing subset numbers to give the Bell numbers, and properly points out that this method, although of pedagogical interest, is quite inefficient as a way of calculating the Bell numbers. I was fascinated, as I always am when I read his articles, by the several much more efficient algorithms he showed for this purpose.