  Ordering permutations by containment of inversion sets yields a fascinating
partial order on the symmetric group: the weak order. This partial order is,
among other things, a semidistributive lattice. As a consequence, every
permutation has a canonical representation as a join of other permutations.
Combinatorially, these canonical join representations can be modeled in terms
of arc diagrams. Moreover, these arc diagrams also serve as a model to
understand quotient lattices of the weak order. A particularly well-behaved
quotient lattice of the weak order is the well-known Tamari lattice, which
appears in many seemingly unrelated areas of mathematics. The arc diagrams
representing the members of the Tamari lattices are better known as noncrossing
partitions. Recently, the Tamari lattices were generalized to parabolic
quotients of the symmetric group. In this article, we undertake a structural
investigation of these parabolic Tamari lattices, and explain how modified arc
diagrams aid the understanding of these lattices.
