Colin Defant
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View article: Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group
Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group Open
For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced b…
View article: Rowmotion and Echelonmotion
Rowmotion and Echelonmotion Open
Given a linear extension $σ$ of a finite poset $R$, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $σ$. This yields a bijection $\mathrm{Ech}_σ\colon R\to R$ that we call e…
View article: Chute Move Posets are Lattices
Chute Move Posets are Lattices Open
For each permutation $w$, we consider the set $\mathrm{PD}(w)$ of reduced pipe dreams for $w$, partially ordered so that cover relations correspond to (generalized) chute moves. Settling a conjecture of Rubey from 2012, we prove that $\mat…
View article: The Affine Tamari Lattice
The Affine Tamari Lattice Open
Given a fixed integer $n\geq 2$, we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattice and a quotient lattice of the cyclic Dyer lattice, which …
View article: Homology in Combinatorial Refraction Billiards
Homology in Combinatorial Refraction Billiards Open
Given a graph $G$ with vertex set $\{1,\ldots,n\}$, we can project the graphical arrangement of $G$ to an $(n-1)$-dimensional torus to obtain a toric hyperplane arrangement. Adams, Defant, and Striker constructed a toric combinatorial refr…
View article: The Pop-Stack Operator on Ornamentation Lattices
The Pop-Stack Operator on Ornamentation Lattices Open
Each rooted plane tree $\mathsf{T}$ has an associated ornamentation lattice $\mathcal{O}(\mathsf{T})$. The ornamentation lattice of an $n$-element chain is the $n$-th Tamari lattice. We study the pop-stack operator $\mathsf{Pop}\colon\math…
View article: 0-Hecke modules, domino tableaux, and type-<i>B</i> quasisymmetric functions
0-Hecke modules, domino tableaux, and type-<i>B</i> quasisymmetric functions Open
We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$ -Hecke algebras. We apply this framework in type B to give…
View article: Torsors and Tilings from Toric Toggling
Torsors and Tilings from Toric Toggling Open
Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a pa…
View article: Bender–Knuth Billiards in Coxeter Groups
Bender–Knuth Billiards in Coxeter Groups Open
Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$ , where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W . If W is of type A , then $\mathscr {L}$ is the set of linear extensions of a poset, and there…
View article: Permutoric promotion: gliding globs, sliding stones, and colliding coins
Permutoric promotion: gliding globs, sliding stones, and colliding coins Open
Defant recently introduced toric promotion, an operator that acts on the labelings of a graph \\(G\\) and serves as a cyclic analogue of Schützenberger's promotion operator. Toric promotion is defined as the composition of certain toggle o…
View article: Boolean, Free, and Classical Cumulants as Tree Enumerations
Boolean, Free, and Classical Cumulants as Tree Enumerations Open
Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called trou…
View article: Foot-Sorting for Socks
Foot-Sorting for Socks Open
If your socks come out of the laundry all mixed up, how should you sort them? We introduce and study a novel foot-sorting algorithm that uses feet to attempt to sort a sock ordering; one can view this algorithm as an analogue of Knuth's st…
View article: Rainbow Stackings of Random Edge-Colorings
Rainbow Stackings of Random Edge-Colorings Open
A rainbow stacking of $r$-edge-colorings $χ_1, \ldots, χ_m$ of the complete graph on $n$ vertices is a way of superimposing $χ_1, \ldots, χ_m$ so that no edges of the same color are superimposed on each other. We determine a sharp threshol…
View article: 0-Hecke Modules, Domino Tableaux, and Type-$B$ Quasisymmetric Functions
0-Hecke Modules, Domino Tableaux, and Type-$B$ Quasisymmetric Functions Open
We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$-Hecke algebras. We apply this framework in type $B$ to giv…
View article: Toric Promotion with Reflections and Refractions
Toric Promotion with Reflections and Refractions Open
Inspired by recent work on refraction billiards in dynamics, we introduce a notion of refraction for combinatorial billiards. This allows us to define a generalization of toric promotion that we call toric promotion with reflections and re…
View article: Tilings of Benzels via Generalized Compression
Tilings of Benzels via Generalized Compression Open
Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on a…
View article: Wiener Indices of Minuscule Lattices
Wiener Indices of Minuscule Lattices Open
The Wiener index of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. …
View article: Bender--Knuth Billiards in Coxeter Groups
Bender--Knuth Billiards in Coxeter Groups Open
Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and th…
View article: Pop-Stack Operators for Torsion Classes and Cambrian Lattices
Pop-Stack Operators for Torsion Classes and Cambrian Lattices Open
The pop-stack operator of a finite lattice $L$ is the map $\mathrm{pop}^{\downarrow}_L\colon L\to L$ that sends each element $x\in L$ to the meet of $\{x\}\cup\text{cov}_L(x)$, where $\text{cov}_L(x)$ is the set of elements covered by $x$ …
View article: Tilings of benzels via the abacus bijection
Tilings of benzels via the abacus bijection Open
Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and pro…
View article: Homomesy via toggleability statistics
Homomesy via toggleability statistics Open
The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies: statistics that have the same average along ev…
View article: Ordering Candidates via Vantage Points
Ordering Candidates via Vantage Points Open
Given an $n$-element set $C\subseteq\mathbb{R}^d$ and a (sufficiently generic) $k$-element multiset $V\subseteq\mathbb{R}^d$, we can order the points in $C$ by ranking each point $c\in C$ according to the sum of the distances from $c$ to t…
View article: Fertilitopes
Fertilitopes Open
We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s . Associated to each permutation $$\pi $$ is a particular set $$\mathcal V(\pi )$$ of integer compositions that…
View article: Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins
Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins Open
The first author recently introduced toric promotion, an operator that acts on the labelings of a graph $G$ and serves as a cyclic analogue of Schützenberger's promotion operator. Toric promotion is defined as the composition of certain to…
View article: Torsors and tilings from toric toggling
Torsors and tilings from toric toggling Open
Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a pa…
View article: Connectedness and cycle spaces of friends-and-strangers graphs
Connectedness and cycle spaces of friends-and-strangers graphs Open
If X = (V (X), E(X)) and Y = (V (Y ), E(Y )) are n-vertex graphs, then their friends-and-strangers graph FS(X, Y ) is the graph whose vertices are the bijections from V (X) to V (Y ) in which two bijections σ and σ are adjacent if and only…
View article: Triangular-grid billiards and plabic graphs
Triangular-grid billiards and plabic graphs Open
Given a polygon \(P\) in the triangular grid, we obtain a permutation \(\pi_P\) via a natural billiards system in which beams of light bounce around inside of \(P\). The different cycles in \(\pi_P\) correspond to the different trajectorie…
View article: The Ungar Games
The Ungar Games Open
Let $L$ be a finite lattice. An Ungar move sends an element $x\in L$ to the meet of $\{x\}\cup T$, where $T$ is a subset of the set of elements covered by $x$. We introduce the following Ungar game. Starting at the top element of $L$, two …
View article: Ungarian Markov Chains
Ungarian Markov Chains Open
We introduce the Ungarian Markov chain ${\bf U}_L$ associated to a finite lattice $L$. The states of this Markov chain are the elements of $L$. When the chain is in a state $x\in L$, it transitions to the meet of $\{x\}\cup T$, where $T$ i…
View article: Semidistrim Lattices
Semidistrim Lattices Open
We introduce semidistrim lattices , a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets …