Partition of a set
Counting partitions

The total number of partitions of an n -element set is the Bell number B. The first several Bell numbers are B = 1, B = 1, B = 2, B = 5, B = 15, B = 52, and B = 203 (sequence A000110 in the OEIS). Bell numbers satisfy the recursion

and have the exponential generating function

The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied from the end of the previous row, and subsequent values are computed by adding two numbers, the number to the left and the number to the above left of the position. The Bell numbers are repeated along both sides of this triangle. The numbers within the triangle count partitions in which a given element is the largest singleton.

The number of partitions of an n -element set into exactly k (non-empty) parts is the Stirling number of the second kind S (n, k).

The number of noncrossing partitions of an n -element set is the Catalan number
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