Stirling numbers c(n,k), S(n,k) of the first and second kind give the answer to two basic counting problems:
-- how many permutations of {1,2,...,n} have k cycles?
-- how many set partitions of {1,2,...,n} have k blocks?
Although these numbers have no simple product formulas, they do have Pascal-triangle-like recursions, and closely related generating functions.
We review this, and then re-interpret this story in terms of Koszul algebras and their Koszul duals, which we also review. The c(n,k) are known to give the Hilbert series for certain well-studied Koszul algebras: the cohomology algebras of configurations of n distinct labeled points in d-space. Alternatively, these are the Orlik-Solomon algebras and graded Varchenko-Gelfand algebras for the reflection hyperplane arrangements of type A. The S(n,k) turn out to be the Hilbert series for their (less-studied) Koszul dual algebras.
One also has representations of the symmetric group on all of these algebras. These sequences of representations turn out to be representation stable in the sense of Church and Farb. Also the Stirling triangle recursions lift to branching rules for these representations. This turns out to be a general feature of supersolvable hyperplane arrangements.
(Joint work with Ayah Almousa and Sheila Sundaram)
