Structure of the Noncrossing Partition Hopf Algebra - Jimi Owen
The Catalan numbers are pervasive in mathematics, counting many different objects. We will study algebraic properties of the set of noncrossing partitions, one member of the Catalan family. In particular we will look at a combinatorial Hopf algebra based on the set of noncrossing partitions. The algebraic structure of Hopf algebras is best studied through certain generators known as primitive elements. We employ the open source mathematics software SAGE to compute these primitive elements in low degrees, and use this data to conjecture a resolution of them in terms of the natural defining basis. We then attempt to understand the combinatorial meaning behind the result.