The Bell numbers count the ways to partition a set of \(n\) elements into non-empty groups: \(B_3 = 5\) because {a,b,c} splits as abc | ab,c | ac,b | bc,a | a,b,c. The sequence: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975.
They satisfy the elegant recurrence \(B_{n+1} = \sum_k \binom{n}{k} B_k\) and are computed by the Bell triangle. Named for Eric Temple Bell — mathematician and, as "John Taine," science-fiction author — though the numbers were studied long before him, including by the medieval Japanese in the Genji-kō incense game, whose 52 patterns are exactly the partitions of 5 elements.