Abstract
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.
Highlights
Given a permutation w of [n] d=ef {1, 2, . . . , n}, an inversion of w is a pair of indices for which the corresponding values of w are out of order
Containment of inversion sets introduces a partial order on the set Sn of all permutations of [n]; the weak order
The diagram of the weak order is the graph on the vertex set Sn in which two permutations are related by an edge if they differ by swapping a descent, i.e. an inversion whose corresponding values are adjacent integers
Summary
Given a permutation w of [n] d=ef {1, 2, . . . , n}, an inversion of w is a pair of indices for which the corresponding values of w are out of order. Each join-irreducible permutation of Sn corresponds to a unique arc connecting two distinct elements of [n], and a certain forcing order on these arcs can be used to characterize quotient lattices of the weak order. One of these quotient lattices is the Tamari lattice, first introduced in [42] via a rotation transformation on binary trees. We conclude this article with an enumerative observation relating the generating function of the Mobius function in the core label order of the parabolic Tamari lattices and the generating function of antichains in certain partially ordered sets in Sect.
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