The Bell number counts all set partitions of an -element finite set, with no restriction on the number of blocks of a set partition. Summing the Stirling numbers of the second kind gives . The first five values, for , are . In particular, . [ref]
https://discrete.openmathbooks.org/more/mdm/sec_adv-partitions.html. Grassl and Levin, More Discrete Mathematics via Graph Theory, definition 3.1.2: Bell numbers
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A Bell number is a number that represents the count of different ways to partition a set into non-empty subsets. More formally, the \( n \)-th Bell number, denoted as \( B_n \), counts the number of ways to partition a set of \( n \) elements. For example: - \( B_0 = 1 \): There is one way to partition an empty set (the empty partition).