Noncrossing Partitions, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Henri Mühle

Published 2018-09-05Version 1

The Tamari lattices and the noncrossing partition lattices are important families of lattices that appear in many seemingly unrelated areas of mathematics, such as group theory, combinatorics, representation theory of the symmetric group, algebraic geometry, and many more. They are also deeply connected on a structural level, since the noncrossing partition lattice can be realized as an alternate way of ordering the ground set of the Tamari lattice. Recently, both the Tamari lattices and the noncrossing partition lattices were generalized to parabolic quotients of the symmetric group. In this article we investigate which structural and enumerative properties survive these generalizations.