Parking Functions as Noncrossing Cosets - From Stanley to Cohen-Macaulay Properties

Explore the mathematical relationship between parking functions and noncrossing partitions in this 42-minute lecture from the Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability." Delve into Stanley's established connection between these concepts, with particular focus on how noncrossing partitions index parking function orbits under symmetric group action. Learn about viewing parking functions as cosets modulo noncrossing subgroups and understand the partial ordering of parking functions through inclusion. Master the Cohen-Macaulay property of this poset and its connection to topological properties of the noncrossing partition lattice. Examine the concept of cluster parking functions, primarily focusing on applications within the symmetric group context while noting the broader applicability to finite Coxeter groups.