Student Combinatorics Seminar

This talk will consist of an overview of an interesting overlap of worlds. A partition λ is called a-core if a does not divide any of the hook-lengths of the Young diagram of shape λ. This simple definition has important connections to geometry, modular representation theory, and the study of lattice paths. For example, the number of partitions that are simultaneously a-core and b-core (with a and b relatively prime) is related to generalized (a,b)-Catalan numbers. The usual q-Catalan numbers are related to Dyck paths by the Major statistic, but an analogue for the major statistic for (a,b)-Dyck paths is still unknown!

One connection that is particularly interesting is to numerical sets - a fancy way to say finite subsets of â„•. These correspond nicely with partitions by using their indicator functions as a guideline for a lattice path. A numerical semigroup is a numerical set that is closed under addition. These turn out to act wonderfully under this correspondence and give us a nice way to construct core partitions. How else do these two worlds interact? We'll try to find out! Speaker(s): Jonathan Gerhard (University of Michigan)