The set of all partitions of a set can be partially ordered by refinement. A partition is a refinement of partition if every subset inside fits inside a subset of . For example, is a refinement of ; but is not because the subset is itself not contained in either subset of . This Demonstration shows the lattice formed by all the sets of partitions of a given set ordered by refinement.

Snapshot 1: The number of partitions at a given level is the Stirling number of the second kind; this figure corresponds to . It follows that the total number of vertices in the graph is the Bell number; in this figure, .

Snapshot 2: For graphs with large numbers of vertices, clearing the "show insets" option can better show the overall structure of the lattice.

Snapshot 3: Other visualizations are available by changing the "graph type" option.

Reference

[1] R. P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge: Cambridge University Press, 1997 pp. 127–128.