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,
.
Browse all topics
