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Art of Computer Programming, Volume 4A, The: Combinatorial Algorithms, Part 1 by Donald E. Knuth

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7.2.1.5. Generating all set partitions

Now let’s shift gears and concentrate on a rather different kind of partition. The partitions of a set are the ways to regard that set as a union of nonempty, disjoint subsets called blocks. For example, we listed the five essentially different partitions of {1, 2, 3} at the beginning of the previous section, in 7.2.1.4–(2) and 7.2.1.4–(4). Those five partitions can also be written more compactly in the form

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using a vertical line to separate one block from another. In this list the elements of each block could have been written in any order, and so could the blocks themselves, because ‘13|2’ and ‘31|2’ and ...

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