## Quick Reference

The number *s*(*n*, *r*) of ways of partitioning a set of *n* elements into *r* cycles. For example, the set {1, 2, 3, 4} can be partitioned into two cycles in the following ways:

So *s*(4, 2)=11. Clearly *s*(*n*, 1)=(*n*−1)! and *s*(*n*, *n*)=1. It can be shown that

*s*(*n*+1, *r*)=*s* (*n*, *r*−1)+*ns*(*n*, *r*)

Some authors define these numbers differently so that they satisfy *s*(*n*+1, *r*)=*s*(*n*, *r*−1)−*n**s*(*n*, *r*). The result is that the values are the same except that some of them occur with a negative sign.

Rather like the binomial coefficients, the Stirling numbers occur as coefficients in certain identities. They are named after the Scottish mathematician James Stirling (1692–1770).

*Subjects:*
Mathematics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.