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papers

Publications (106)

math.NT2017

Revisiting The Riemann Zeta Function at Positive Even Integers

Krishnaswami Alladi, Colin Defant

Using Parseval's identity for the Fourier coefficients of $x^{k}$ , we provide a new proof that $I^¶ (2 k) = \frac{( - 1 ) ^{k + 1} B _{2 k} ( 2 I ¨ ) ^{2 k}}{2 ( 2 k )!}$ .

math.CO2024

0-Hecke Modules, Domino Tableaux, and Type- $B$ Quasisymmetric Functions

Colin Defant, Dominic Searles

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 $0$ …

math.CO2023

Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins

Colin Defant, Rachana Madhukara, Hugh Thomas

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 SchÃ¼tzenberger's promotion operator.…

math.CO2015

An Anti-Ramsey Problem Concerning Complete Bipartite Graphs

Stephan Cho, Jay Cummings, Colin Defant +1

We consider quadruples of positive integers $(a, b, m, n)$ with $a \leq b$ and $m \leq n$ such that any proper edge-coloring of the complete bipartite graph $K_{m, n}$ contains a rainbow…

math.CO2021

Symmetry of Narayana numbers and rowvacuation of root posets

Colin Defant, Sam Hopkins

For a Weyl group $W$ of rank $r$ , the $W$ -Catalan number is the number of antichains of the poset of positive roots, and the $W$ -Narayana numbers refine the $W$ -Catalan number by k…

math.CO2018

Stack-sorting for Words

Colin Defant, Noah Kravitz

We introduce operators $hare$ and $tortoise$ , which act on words as natural generalizations of West's stack-sorting map. We show that the heuristically slower alg…

math.CO2025

The Affine Tamari Lattice

Grant Barkley, Colin Defant

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 sublattic…

math.CO2020

Fertility Monotonicity and Average Complexity of the Stack-Sorting Map

Colin Defant

Let $D_{n}$ denote the average number of iterations of West's stack-sorting map $s$ that are needed to sort a permutation in $S_{n}$ into the identity permutation $123\cdots n…

math.CO2026

Sets with Few Subset Sums

Ruben Carpenter, Colin Defant, Noah Kravitz

It is a classical fact that every $n$ -element set of positive reals has at least $(2 n + 1) + 1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressi…

math.CO2023

Extensions of Hitomezashi Patterns

Colin Defant, Noah Kravitz, Bridget Eileen Tenner

Hitomezashi, a form of traditional Japanese embroidery, gives rise to intricate arrangements of axis-parallel unit-length stitches in the plane. Pete studied these patterns in the…

math.CO2022

Stack-Sorting for Coxeter Groups

Colin Defant

Given an essential semilattice congruence $\equiv$ on the left weak order of a Coxeter group $W$ , we define the Coxeter stack-sorting operator $S_{\equiv} : W \to W$ by ${\bf S}_\…

math.CO2021

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

Colin Defant

We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$ . We fi…

math.CO2021

Meeting Covered Elements in $I^½$ -Tamari Lattices

Colin Defant

For each complete meet-semilattice $M$ , we define an operator $Pop_{M} : M \to M$ by \[\mathsf{Pop}_M(x)=\bigwedge(\{y\in M:y\lessdot x\}\cup\{x\}).\] When $M$ is the right wea…

math.CO2026

Short Proofs in Algebraic and Enumerative Combinatorics

Colin Defant

We present several short proofs that resolve open problems from the algebraic and enumerative combinatorics literature. First, we consider the echelonmotion operator on modular lat…

math.CO2018

Enumerating Cliques in Direct Product Graphs

Colin Defant

The unitary Cayley graph of $Z / n Z$ , denoted $G_{Z / n Z}$ , is the graph with vertices $0, 1, \dots,$ $n - 1$ in which two vertices are adjacent if and onl…

math.NT2018

Connected Components of Complex Divisor Functions

Colin Defant

For any complex number $c$ , define the divisor function $I ¨ _{c} : N \to C$ by $I ¨ _{c} (n) = d ∣ n \sum d^{c}$ . Let $I ¨ _{c} (N)$ denote the t…

math.CO2024

Bender--Knuth Billiards in Coxeter Groups

Grant Barkley, Colin Defant, Eliot Hodges +2

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 $L$ of $W$ . If $W$ is of type $A$ , then $\mat…

math.PR2024

Random Subwords and Pipe Dreams

Colin Defant

Fix a probability $p \in (0, 1)$ . Let $s_{i}$ denote the transposition in the symmetric group $S_{n}$ that swaps $i$ and $i + 1$ . Given a word $w$ over the alphabet $\{s…

math.CO2024

The Ungar Games

Colin Defant, Noah Kravitz, Nathan Williams

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 fol…

math.CO2019

On Anti-Powers in Aperiodic Recurrent Words

Aaron Berger, Colin Defant

Fici, Restivo, Silva, and Zamboni define a $\textit{$k$-anti-power}$ to be a concatenation of $k$ consecutive words that are pairwise distinct and have the same length. They ask fo…

math.CO2018

Flexible Toggles and Symmetric Invertible Asynchronous Elementary Cellular Automata

Colin Defant

A sequential dynamical system (SDS) consists of a graph $G$ with vertices $v_{1}, v_{2}, \dots, v_{n}$ , a state set $A$ , a collection of "vertex functions" ${f_{v_{i}}}_{i = 1}^{n}$ , and a per…

math.CO2019

Stack-Sorting Preimages of Permutation Classes

Colin Defant

We extend and generalize many of the enumerative results concerning West's stack-sorting map $s$ . First, we prove a useful theorem that allows one to efficiently compute $|s^{-1}(Ï…

math.CO2022

Pop-Stack-Sorting for Coxeter Groups

Colin Defant

Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $Pop : W \to W$ to be the map that fixes the identity element and sends each noniden…

math.NT2015

On the Density of Ranges of Generalized Divisor Functions

Colin Defant

The range of the divisor function $I ¨ _{- 1}$ is dense in the interval $[1, \infty)$ . However, the range of the function $I ¨ _{- 2}$ is not dense in the interval $\displaystyle{\left[1,…

math.CO2023

Ungarian Markov Chains

Colin Defant, Rupert Li

We introduce the Ungarian Markov chain $U_{L}$ 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…

math.CO2025

Homology in Combinatorial Refraction Billiards

Colin Defant, Derek Liu

Given a graph $G$ with vertex set ${1, \dots, n}$ , we can project the graphical arrangement of $G$ to an $(n - 1)$ -dimensional torus to obtain a toric hyperplane arrangement. Adams,…

math.CO2024

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…

math.CO2020

Asymptotics of 3-stack-sortable permutations

Colin Defant, Andrew Elvey Price, Anthony J Guttmann

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we exte…

math.NT2018

Ranges of Unitary Divisor Functions

Colin Defant

For any real $t$ , the unitary divisor function $I ¨ _{t}^{*}$ is the multiplicative arithmetic function defined by $I ¨ _{t}^{*} (p^{I^} \pm) = 1 + p^{I^\pm t}$ for all primes $p$ and positive integers $I^\pm$ .…

math.CO2024

Tilings of Benzels via Generalized Compression

Colin Defant, Leigh Foster, Rupert Li +2

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 protot…

math.CO2017

Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

Colin Defant

Let $[K_{n}, f, I ¨ ]$ be the (global) SDS map of a sequential dynamical system (SDS) defined over the complete graph $K_{n}$ using the update order $I ¨ \in S_{n}$ in which all vertex functio…

math.PR2025

Random Subwords and Billiard Walks in Affine Weyl Groups

Colin Defant, Pakawut Jiradilok, Elchanan Mossel

Let $W$ be an irreducible affine Weyl group, and let $b$ be a finite word over the alphabet of simple reflections of $W$ . Fix a probability $p \in (0, 1)$ . For each integer $…

math.PR2026

Random Combinatorial Billiards and Stoned Exclusion Processes

Colin Defant

We introduce and study several random combinatorial billiard trajectories. Such a system, which depends on a fixed parameter $p \in (0, 1)$ , models a beam of light that travels in a E…

math.NT2018

On the genus of a quotient of a numerical semigroup

Ayomikun Adeniran, Steve Butler, Colin Defant +6

We find a relation between the genus of a quotient of a numerical semigroup $S$ and the genus of $S$ itself. We use this identity to compute the genus of a quotient of $S$ when $S$ …

math.CO2020

Catalan Intervals and Uniquely Sorted Permutations

Colin Defant

For each positive integer $k$ , we consider five well-studied posets defined on the set of Dyck paths of semilength $k$ . We prove that uniquely sorted permutations avoiding various…

math.CO2024

Mixing on Generalized Associahedra

William Chang, Colin Defant, Daniel Frishberg

Eppstein and Frishberg recently proved that the mixing time for the simple random walk on the $1$ -skeleton of the associahedron is $O (n^{3} lo g^{3} n)$ . We obtain similar rapid mixing…

math.CO2023

Rowmotion Markov Chains

Colin Defant, Rupert Li, Evita Nestoridi

Rowmotion is a certain well-studied bijective operator on the distributive lattice $J (P)$ of order ideals of a finite poset $P$ . We introduce the rowmotion Markov chain ${\bf M}_{J…

math.CO2020

Pattern-Avoiding Permutation Powers

Amanda Burcroff, Colin Defant

Recently, BÃ³na and Smith defined strong pattern avoidance, saying that a permutation $I ¨ $ strongly avoids a pattern $I ¨ $ if $I ¨ $ and $I ¨ ^{2}$ both avoid $I ¨ $ . They conjectured that…

math.CO2023

Homomesy via Toggleability Statistics

Colin Defant, Sam Hopkins, Svetlana PoznanoviÄ +1

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…

math.CO2025

The Pop-Stack Operator on Ornamentation Lattices

Khalid Ajran, Colin Defant

Each rooted plane tree $T$ has an associated ornamentation lattice $O (T)$ . The ornamentation lattice of an $n$ -element chain is the $n$ -th Tamari lattic…

math.CO2024

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…

math.CO2020

Quantifying Noninvertibility in Discrete Dynamical Systems

Colin Defant, James Propp

Given a finite set $X$ and a function $f : X \to X$ , we define the degree of noninvertibility of $f$ to be $deg (f) = \frac{1}{∣ X ∣} x \in X \sum ∣ f^{- 1} (f (x)) ∣$ . This…

math.CO2020

Supertrees

Colin Defant, Noah Kravitz, Ashwin Sah

A $k$ -universal permutation, or $k$ -superpermutation, is a permutation that contains all permutations of length $k$ as patterns. The problem of finding the minimum length of a $k$ -…

math.NT2014

On Sparsely Schemmel Totient Numbers

Colin Defant

For each positive integer $r$ , let $S_{r}$ denote the $r^{t h}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^Î±)=\begin{cases} 0, & \mbox{if } p\…

math.NT2015

On Ranges of Variants of the Divisor Functions that are Dense

Colin Defant

For a real number $t$ , let $s_{t}$ be the multiplicative arithmetic function defined by $s_{t} (p^{I^} \pm) = j = 0 \sum I^\pm (- p^{t})^{j}$ for all primes $p$ and positive integers $I^\pm$ …

math.CO2015

Upper Bounds for Stern's Diatomic Sequence and Related Sequences

Colin Defant

Let $(s_{2} (n))_{n = 0}^{\infty}$ denote Stern's diatomic sequence. For $n \geq 2$ , we may view $s_{2} (n)$ as the number of partitions of $n - 1$ into powers of $2$ with each part occurring a…

math.DS2026

The Minary Primitive of Computational Autopoiesis

Daniel Connor, Colin Defant

We introduce Minary, a computational framework designed as a candidate for the first formally provable autopoietic primitive. Minary represents interacting probabilistic events as…

math.CO2021

Typical and Extremal Aspects of Friends-and-Strangers Graphs

Noga Alon, Colin Defant, Noah Kravitz

Given graphs $X$ and $Y$ with vertex sets $V (X)$ and $V (Y)$ of the same cardinality, the friends-and-strangers graph $FS (X, Y)$ is the graph whose vertex set consists of al…

math.CO2024

Pop, Crackle, Snap (and Pow): Some Facets of Shards

Colin Defant, Nathan Williams

Reading cut the hyperplanes in a real central arrangement $H$ into pieces called \emph{shards}, which reflect order-theoretic properties of the arrangement. We show that s…

math.CO2025

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…

math.CO2022

Loops and Regions in Hitomezashi Patterns

Colin Defant, Noah Kravitz

Hitomezashi patterns, which originate from traditional Japanese embroidery, are intricate arrangements of unit-length line segments called stitches. The stitches connect to form hi…

math.NT2015

An Extension of the Abundancy Index to Certain Quadratic Rings

Colin Defant

We begin by introducing an extension of the traditional abundancy index to imaginary quadratic rings with unique factorization. After showing that many of the properties of the tra…

math.CO2023

Ordering Candidates via Vantage Points

Noga Alon, Colin Defant, Noah Kravitz +1

Given an $n$ -element set $C \subseteq R^{d}$ and a (sufficiently generic) $k$ -element multiset $V \subseteq R^{d}$ , we can order the points in $C$ by ranking each point $…

math.CO2020

Polyurethane Toggles

Colin Defant

We consider the involutions known as "toggles," which have been used to give simplified proofs of the fundamental properties of the promotion and evacuation maps. We transfer these…

math.CO2020

Promotion Sorting

Colin Defant, Noah Kravitz

SchÃ¼tzenberger's promotion operator is an extensively-studied bijection that permutes the linear extensions of a finite poset. We introduce a natural extension $\partial$ of this…

math.CO2020

Fertility, Strong Fertility, and Postorder Wilf Equivalence

Colin Defant

We introduce "fertility Wilf equivalence," "strong fertility Wilf equivalence," and "postorder Wilf equivalence," three variants of Wilf equivalence for permutation classes that fo…

math.NT2014

Unitary Multiperfect Numbers in Certain Quadratic Rings

Colin Defant

A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\frac{n}{c}$ . For any integer $k$ , the function $I ¨ _{k}^{*}$ is…

math.CO2025

Chute Move Posets are Lattices

Ilani Axelrod-Freed, Colin Defant, Hanna Mularczyk +2

For each permutation $w$ , we consider the set $PD (w)$ of reduced pipe dreams for $w$ , partially ordered so that cover relations correspond to (generalized) chute moves. Se…

math.NT2015

Multiperfect Numbers in Certain Quadratic Rings

Colin Defant

Using an extension of the abundancy index to imaginary quadratic rings that are unique factorization domains, we investigate what we call $n$ -powerfully $t$ -perfect numbers in thes…

math.CO2023

Pop-Stack Operators for Torsion Classes and Cambrian Lattices

Emily Barnard, Colin Defant, Eric J. Hanson

The pop-stack operator of a finite lattice $L$ is the map $pop_{L}^{↓} : L \to L$ that sends each element $x \in L$ to the meet of ${x} \cup cov_{L} (x)$ , whe…

math.CO2021

Semidistrim Lattices

Colin Defant, Nathan Williams

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…

math.CO2018

Domination and Upper Domination of Direct Product Graphs

Colin Defant, Sumun Iyer

The unitary Cayley graph of $Z / n Z$ , denoted $X_{Z / n Z}$ , has vertices $0, 1, \dots, n - 1$ with $x$ adjacent to $y$ if $x - y$ is relatively prim…

math.CO2026

On the number of permutation-twisted dot products

Ruben Carpenter, Colin Defant, Noah Kravitz

Let $K$ be a field of characteristic $0$ . For each choice of distinct $a_{1}, \dots, a_{n} \in K$ and distinct $b_{1}, \dots, b_{n} \in K$ , consider the sum $S=\…

math.CO2022

Troupes, Cumulants, and Stack-Sorting

Colin Defant

Several sequences of free cumulants that count binary plane trees correspond to sequences of classical cumulants that count the decreasing versions of the same trees. Using two new…

math.CO2023

Fertilitopes

Colin Defant

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West's stack-sorting map $s$ . Associated to each permutation $I ¨ $ is a particular set $…

math.CO2019

Proofs of Conjectures about Pattern-Avoiding Linear Extensions

Colin Defant

After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consi…

math.CO2021

Friends and Strangers Walking on Graphs

Colin Defant, Noah Kravitz

Given graphs $X$ and $Y$ with vertex sets $V (X)$ and $V (Y)$ of the same cardinality, we define a graph $FS (X, Y)$ whose vertex set consists of all bijections $Ï:V(X)\to V(…

math.CO2019

Descents in $t$ -Sorted Permutations

Colin Defant

Let $s$ denote West's stack-sorting map. A permutation is called $t - sorted$ if it is of the form $s^{t} (I^¼)$ for some permutation $I^¼$ . We prove that the maximum number of d…

math.CO2022

Tilings of Benzels via the Abacus Bijection

Colin Defant, Rupert Li, James Propp +1

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…

math.CO2025

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group

Colin Defant, Melissa Sherman-Bennett, Nathan Williams

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 $W$ -permutahed…

math.CO2017

Poset Pattern-Avoidance Problems Posed by Yakoubov

Colin Defant

Extending the work of Yakoubov, we enumerate the linear extensions of comb posets that avoid certain length- $3$ patterns. We resolve many of Yakoubov's open problems and prove both…

math.NT2015

A Note about Iterated Arithmetic Functions

Colin Defant

Let $f : N \to N_{0}$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $I^\pm$ , $f (p^{I^} \pm) < p^{I^} \pm$ and $f(p)\vert f(p^Î±…

math.NT2014

On Schemmel Nontotient Numbers

Colin Defant

For each positive integer $r$ , let $S_{r}$ denote the $r^{t h}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^Î±)=\begin{cases} 0, & \mbox{if } p\…

math.NT2014

On 2-powerfully Perfect Numbers in Three Quadratic Rings

Colin Defant

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$ -powerfully perfect numbers in these rings. This definit…

math.CO2020

Counting 3-Stack-Sortable Permutations

Colin Defant

We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map $s$ . As a first application, we give a new proof o…

math.CO2016

Anti-Power Prefixes of the Thue-Morse Word

Colin Defant

Recently, Fici, Restivo, Silva, and Zamboni defined a $k$ -anti-power to be a word of the form $w_{1} w_{2} \dots w_{k}$ , where $w_{1}, w_{2}, \dots, w_{k}$ are distinct words of the same length.…

math.CO2018

Preimages under the Stack-Sorting Algorithm

Colin Defant

We use a method for determining the number of preimages of any permutation under the stack-sorting map in order to obtain recursive upper bounds for the numbers $W_{t} (n)$ and $W_t(n…

math.CO2024

Rainbow Stackings of Random Edge-Colorings

Noga Alon, Colin Defant, Noah Kravitz

A rainbow stacking of $r$ -edge-colorings $I ¨ _{1}, \dots, I ¨ _{m}$ of the complete graph on $n$ vertices is a way of superimposing $I ¨ _{1}, \dots, I ¨ _{m}$ so that no edges of the same colo…

math.CO2023

Torsors and tilings from toric toggling

Colin Defant, Michael Joseph, Matthew Macauley +1

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 s…

math.CO2020

Highly Sorted Permutations and Bell Numbers

Colin Defant

Let $s$ denote West's stack-sorting map. For all positive integers $m$ and all integers $n \geq 2 m - 2$ , we give a simple characterization of the set $s^{n - m} (S_{n})$ ; as a consequence,…

math.CO2020

Stack-Sorting with Consecutive-Pattern-Avoiding Stacks

Colin Defant, Kai Zheng

We introduce consecutive-pattern-avoiding stack-sorting maps $SC_{I ¨} $ , which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classica…

math.CO2020

Stack-Sorting, Set Partitions, and Lassalle's Sequence

Colin Defant, Michael Engen, Jordan A. Miller

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to s…

math.CO2025

Rowmotion and Echelonmotion

Colin Defant, Yuhan Jiang, Rene Marczinzik +4

Given a linear extension $I ¨ $ of a finite poset $R$ , we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $I ¨ $ . This yi…

math.CO2017

Unitary Cayley Graphs of Dedekind Domain Quotients

Colin Defant

If $X$ is a commutative ring with unity, then the unitary Cayley graph of $X$ , denoted $G_{X}$ , is defined to be the graph whose vertex set is $X$ and whose edge set is $\{\{a,b\}\co…

math.CO2023

Wiener Indices of Minuscule Lattices

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…

math.CO2018

Postorder Preimages

Colin Defant

Given a set $Y$ of decreasing plane trees and a permutation $I ¨ $ , how many trees in $Y$ have $I ¨ $ as their postorder? Using combinatorial and geometric constructions, we provide a…

math.CO2023

Triangular-Grid Billiards and Plabic Graphs

Colin Defant, Pakawut Jiradilok

Given a polygon $P$ in the triangular grid, we obtain a permutation $I ¨ _{P}$ via a natural billiards system in which beams of light bounce around inside of $P$ . The different cycles…

math.CO2021

Crystal Pop-Stack Sorting and Type A Crystal Lattices

Colin Defant, Nathan Williams

Given a complex simple Lie algebra $g$ and a dominant weight $I^»$ , let $B_{I^} »$ be the crystal poset associated to the irreducible representation of $g$ …

math.NT2017

On the Density of Ranges of Generalized Divisor Functions with Restricted Domains

Colin Defant

We begin by defining functions $I ¨ _{t, k}$ , which are generalized divisor functions with restricted domains. For each positive integer $k$ , we show that, for $r > 1$ , the range of $Ï…

math.CO2021

Coxeter Pop-Tsack Torsing

Colin Defant, Nathan Williams

Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$ , we define the Coxeter pop-tsack torsing operator $Pop_{T} : W \to W$ by $\mathsf{Pop}_T(w)=w\cdot…

math.NT2015

On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

Colin Defant

We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function…

math.CO2022

Connectedness and Cycle Spaces of Friends-and-Strangers Graphs

Colin Defant, David Dong, Alan Lee +1

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 $V (X)$ t…

math.CO2026

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…

math.CO2024

Rowmotion on $m$ -Tamari and BiCambrian Lattices

Colin Defant, James Lin

Thomas and Williams conjectured that rowmotion acting on the rational $(a, b)$ -Tamari lattice has order $a + b - 1$ . We construct an equivariant bijection that proves this conjecture wh…

math.CO2023

Motzkin Intervals and Valid Hook Configurations

Colin Defant

We define a new natural partial order on Motzkin paths that serves as an intermediate step between two previously-studied partial orders. We provide a bijection between valid hook…

math.PR2025

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…

math.CO2021

The runsort permuton

Noga Alon, Colin Defant, Noah Kravitz

Suppose we choose a permutation $I ¨ $ uniformly at random from $S_{n}$ . Let $runsort (I ¨ )$ be the permutation obtained by sorting the ascending runs of $I ¨ $ into lexicographic…

math.PR2025

Permutons from Demazure Products

Colin Defant

We construct and analyze several new families of permutons arising from random processes involving the Demazure product on the symmetric group. First, we consider Demazure products…

math.NT2018

Complex Divisor Functions

Colin Defant

For any complex number $c$ , let $I ¨ _{c} : N \to C$ denote the divisor function defined by $I ¨ _{c} (n) = d ∣ n \sum d^{c}$ for all $n \in N$ , and…

math.CO2026

Toric Promotion with Reflections and Refractions

Ashleigh Adams, Colin Defant, Jessica Striker

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 pro…

Publications (106)

Revisiting The Riemann Zeta Function at Positive Even Integers

0-Hecke Modules, Domino Tableaux, and Type-B Quasisymmetric Functions

Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins

An Anti-Ramsey Problem Concerning Complete Bipartite Graphs

Symmetry of Narayana numbers and rowvacuation of root posets

Stack-sorting for Words

The Affine Tamari Lattice

Fertility Monotonicity and Average Complexity of the Stack-Sorting Map

Sets with Few Subset Sums

Extensions of Hitomezashi Patterns

Stack-Sorting for Coxeter Groups

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

Meeting Covered Elements in I^½-Tamari Lattices

Short Proofs in Algebraic and Enumerative Combinatorics

Enumerating Cliques in Direct Product Graphs

Connected Components of Complex Divisor Functions

Bender--Knuth Billiards in Coxeter Groups

Random Subwords and Pipe Dreams

The Ungar Games

On Anti-Powers in Aperiodic Recurrent Words

Flexible Toggles and Symmetric Invertible Asynchronous Elementary Cellular Automata

Stack-Sorting Preimages of Permutation Classes

Pop-Stack-Sorting for Coxeter Groups

On the Density of Ranges of Generalized Divisor Functions

Ungarian Markov Chains

Homology in Combinatorial Refraction Billiards

Boolean, Free, and Classical Cumulants as Tree Enumerations

Asymptotics of 3-stack-sortable permutations

Ranges of Unitary Divisor Functions

Tilings of Benzels via Generalized Compression

Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

Random Subwords and Billiard Walks in Affine Weyl Groups

Random Combinatorial Billiards and Stoned Exclusion Processes

On the genus of a quotient of a numerical semigroup

Catalan Intervals and Uniquely Sorted Permutations

Mixing on Generalized Associahedra

Rowmotion Markov Chains

Pattern-Avoiding Permutation Powers

Homomesy via Toggleability Statistics

The Pop-Stack Operator on Ornamentation Lattices

Operahedron Lattices

Quantifying Noninvertibility in Discrete Dynamical Systems

Supertrees

On Sparsely Schemmel Totient Numbers

On Ranges of Variants of the Divisor Functions that are Dense

Upper Bounds for Stern's Diatomic Sequence and Related Sequences

The Minary Primitive of Computational Autopoiesis

Typical and Extremal Aspects of Friends-and-Strangers Graphs

Pop, Crackle, Snap (and Pow): Some Facets of Shards

Extended Weak Order for the Rank 3 Universal Coxeter Group

Loops and Regions in Hitomezashi Patterns

An Extension of the Abundancy Index to Certain Quadratic Rings

Ordering Candidates via Vantage Points

Polyurethane Toggles

Promotion Sorting

Fertility, Strong Fertility, and Postorder Wilf Equivalence

Unitary Multiperfect Numbers in Certain Quadratic Rings

Chute Move Posets are Lattices

Multiperfect Numbers in Certain Quadratic Rings

Pop-Stack Operators for Torsion Classes and Cambrian Lattices

Semidistrim Lattices

Domination and Upper Domination of Direct Product Graphs

On the number of permutation-twisted dot products

Troupes, Cumulants, and Stack-Sorting

Fertilitopes

Proofs of Conjectures about Pattern-Avoiding Linear Extensions

Friends and Strangers Walking on Graphs

Descents in t-Sorted Permutations

Tilings of Benzels via the Abacus Bijection

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group

Poset Pattern-Avoidance Problems Posed by Yakoubov

A Note about Iterated Arithmetic Functions

On Schemmel Nontotient Numbers

On 2-powerfully Perfect Numbers in Three Quadratic Rings

Counting 3-Stack-Sortable Permutations

Anti-Power Prefixes of the Thue-Morse Word

Preimages under the Stack-Sorting Algorithm

Rainbow Stackings of Random Edge-Colorings

Torsors and tilings from toric toggling

0-Hecke Modules, Domino Tableaux, and Type- $B$ Quasisymmetric Functions

Meeting Covered Elements in $I^½$ -Tamari Lattices

Descents in $t$ -Sorted Permutations

Rowmotion on $m$ -Tamari and BiCambrian Lattices

Reduced Random Walks in the Hyperbolic Plane $! ?$