Noncrossing (set) partitions of an n-element set constitute a fascinating subset of all set partitions with surprising combinatorial properties. Among other things they belong to the family of Catalan objects, which means that their cardinality is given by the nth Catalan number.
Besides several beautiful combinatorial properties—for instance they constitute a self-dual lattice under (dual) refinement order—noncrossing partitions appear in various mathematical contexts. They can be used for instance to describe free cumulants of a noncommutative probability distribution, and Möbius Inversion on the noncrossing partition lattice allows for enumerating connected positroids. Moreover, they are naturally in bijection with families of pairwise orthogonal indecomposable representations of a path-graph.
It is the goal of this class to achieve a basic introduction into the theory of noncrossing partitions, where the focus lies on different combinatorial methods. The participants are expected to participate actively, and work out particular topics on their own, and present them in class.
The preliminary script is here. Another (German) script on posets and lattices is here. The class takes place as follows:
- Wednesday, 13:00–14:30 @ WIL/A124,
- Thursday, 11:10–12:40 @ SE2/0102/U.
Further details will be discussed during the first meeting on April 05, 2017.