# Bell Numbers

defined as
$\displaystyle B(n) = \sum_{k=0}^n \left\{ {n \atop k}\right\}$
counts the total number of ways to partition n distinct objects into disjoint subsets (or blocks) where
$\displaystyle \left\{ {n \atop k}\right\}$
is the well known Stirling numbers of the second kind.

The most famous Bell number is probably B(5) = 52 because of its graphical appearance in the japanese literature the Tale of Genji. This was noted in both Richard Stanley’s book as well as the wikipedia page for bell numbers. I went to the Central Library to check out the book. There were about 10 or so related books, including a couple of English translations, guide books, perspectives. Trust the literary people to completely leave out the chapter numbering. The best reference is of course Martin Gardner’s column in Scientific American, which includes actual references to the appearing of the numbering in the literature.

In another remarkable coincidence, these symbols made an appearance in the colloquium talk in the department on 23 Feb. Apparently, S Ramanujan studied the Bell numbers before Bell!

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