Non-crossing partitions are combinatorial objects appearing in close connection with the following structures:

• Coxeter groups,
• braid groups,
• Catalan numbers,
• associahedra
and many others...

All non-crossing partitions of the same set together form a poset allowing some nice duality. The structure of non-crossing partitions appears in various contexts, but it is still an open question how to extend the notion to other situations so that it would still preserve its nice properties.

Prerequisites: Algebra II.
Also, Topics in Combinatorics IV is recommended as a corequisite, but not required.

Resources: You can find some sources here.