Stirling numbers of the second kind

The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources.

Analogously, the ordered Bell numbers can be computed from the Stirling numbers of the second kind via

More directly, let two sets contain positions of 1's in binary representations of results of respective expressions:

Another explicit expansion of the recurrence-relation gives identities in the spirit of the above example.

The Stirling numbers of the second kind are given by the explicit formula:

A mixed bivariate generating function for the Stirling numbers of the second kind is