12 papers
Invariance of rowmotion for variants of the Tamari lattice
Ben Adenbaum, Emily Barnard, Cesar Ceballos +6
We show that the rowmotion operator from dynamical algebraic combinatorics behaves the same on all alt -Tamari lattices for a fixed lattice path . We use this invariance of r…
Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants
Colin Defant, Nathan Williams
Much work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combina…
Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group
Colin Defant, Melissa Sherman-Bennett, Nathan Williams
For each finite Coxeter group and each standard Coxeter element of , we construct a triangulation of the -permutahedron. For particular realizations of the -permutahed…
Extended Weak Order for the Rank 3 Universal Coxeter Group
Grant Barkley, Colin Defant, Patricia Hersh +3
The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by pr…
Rowmotion and Echelonmotion
Colin Defant, Yuhan Jiang, Rene Marczinzik +4
Given a linear extension of a finite poset , we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of with respect to . This yiel…
Chute Move Posets are Lattices
Ilani Axelrod-Freed, Colin Defant, Hanna Mularczyk +2
For each permutation , we consider the set of reduced pipe dreams for , partially ordered so that cover relations correspond to (generalized) chute moves. Se…
Reduced Random Walks in the Hyperbolic Plane
Colin Defant, Mitchell Lee
We study Lam's reduced random walk in a hyperbolic triangle group, which we view as a random walk in the upper half-plane. We prove that this walk converges almost surely to a poin…
The Affine Tamari Lattice
Grant Barkley, Colin Defant
Given a fixed integer , we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattic…
The Pop-Stack Operator on Ornamentation Lattices
Khalid Ajran, Colin Defant
Each rooted plane tree has an associated ornamentation lattice . The ornamentation lattice of an -element chain is the -th Tamari lattic…
Boolean, Free, and Classical Cumulants as Tree Enumerations
Colin Defant, Mitchell Lee
Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatori…
Operahedron Lattices
Colin Defant, Andrew Sack
Laplante-Anfossi associated to each rooted plane tree a polytope called an operahedron. He also defined a partial order on the vertex set of an operahedron and asked if the resulti…
Pop, Crackle, Snap (and Pow): Some Facets of Shards
Colin Defant, Nathan Williams
Reading cut the hyperplanes in a real central arrangement into pieces called \emph{shards}, which reflect order-theoretic properties of the arrangement. We show that s…