Stirling's Triangles

Ignoring signs, Stirling numbers of the first kind count the number of permutations of that have cycles.

Stirling numbers of the second kind count the number of ways the set can be partitioned into an unordered family of nonempty subsets. The sums of the columns are the Bell numbers , which count the number of set partitions of a set of elements.

It is remarkable that the two types of triangles are inverses of each other as infinite triangular matrices.

Stirling numbers of the first kind satisfy the recurrence , while those of the second kind satisfy the very similar . The recurrences are also similar to the simpler recurrence formula for the binomial coefficients, .