arXiv subjects

Colin Defant

Publications and source records attributed to Colin Defant.

At least 37 records · Page 2Linked to original sources

arXiv2025

Random Subwords and Billiard Walks in Affine Weyl Groups

Let $W$ be an irreducible affine Weyl group, and let $\mathsf{b}$ be a finite word over the alphabet of simple reflections of $W$. Fix a probability $p\in(0,1)$. For each integer $K\geq 0$, let $\mathsf{sub}_p(\mathsf{b}^K)$ be the random subword of $\mathsf{b}^K$ obtained by deleting each letter independently with probability $1-p$. Let $v_p(\mathsf{b}^K)$ be the element of $W$ represented by $\mathsf{sub}_p(\mathsf{b}^K)$. One can view $v_p(\mathsf{b}^K)$ geometrically as a random alcove; in many cases, this alcove can be seen as the location after a certain amount of time of a random billiard trajectory that, upon hitting a hyperplane in the Coxeter arrangement of $W$, reflects off of the hyperplane with probability $1-p$. We show that the asymptotic distribution of $v_p(\mathsf{b}^K)$ is a central spherical multivariate normal distribution with some variance $σ_{\mathsf{b}}^2$ depending on $\mathsf{b}$ and $p$. We provide a formula to compute $σ_{\mathsf{b}}^2$ that is remarkably simple when $\mathsf{b}$ contains only one occurrence of the simple reflection that is not in the associated finite Weyl group. As a corollary, we provide an asymptotic formula for $\mathbb{E}[\ell(v_p(\mathsf{b}^K))]$, the expected Coxeter length of $v_p(\mathsf{b}^K)$. For example, when $W=\widetilde A_{r}$ and $\mathsf{b}$ contains each simple reflection exactly once, we find that \[\lim_{K\to\infty}\frac{1}{\sqrt{K}}\mathbb{E}[\ell(v_p(\mathsf{b}^K))]=\sqrt{\frac{2}πr(r+1)\frac{p}{1-p}}.\]

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arXiv2025

The Pop-Stack Operator on Ornamentation Lattices

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 $\mathsf{Pop}\colon\mathcal{O}(\mathsf{T})\to\mathcal{O}(\mathsf{T})$, which sends each element $δ$ to the meet of the elements covered by or equal to $δ$. We compute the maximum size of a forward orbit of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Defant for Tamari lattices. We also characterize the image of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Hong for Tamari lattices. For each integer $k\geq 0$, we provide necessary conditions for an element of $\mathcal{O}(\mathsf{T})$ to be in the image of $\mathsf{Pop}^k$. This allows us to completely characterize the image of $\mathsf{Pop}^k$ on a Tamari lattice.

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arXiv2024

Bender--Knuth Billiards in Coxeter Groups

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 are important Bender--Knuth involutions $\mathrm{BK}_i\colon\mathscr{L}\to\mathscr{L}$ indexed by elements of $I$. For arbitrary $W$ and for each $i\in I$, we introduce an operator $τ_i\colon W\to W$ (depending on $\mathscr{L}$) that we call a noninvertible Bender--Knuth toggle; this operator restricts to an involution on $\mathscr{L}$ that coincides with $\mathrm{BK}_i$ in type $A$. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm{Pro}_c=τ_{i_n}\cdotsτ_{i_1}$. We say $W$ is futuristic if for every nonempty finite convex set $\mathscr{L}$, every Coxeter element $c$, and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm{Pro}_c^K(u)\in\mathscr{L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When $W$ is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of $W$, then $τ_{i_N}\cdotsτ_{i_1}(W)=\mathscr{L}$; this allows us to determine the smallest integer $\mathrm{M}(c)$ such that $\mathrm{Pro}_c^{\mathrm{M}(c)}(W)=\mathscr{L}$ for all $\mathscr{L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.

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arXiv2024

Boolean, Free, and Classical Cumulants as Tree Enumerations

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 troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.

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arXiv2024

Mixing on Generalized Associahedra

Eppstein and Frishberg recently proved that the mixing time for the simple random walk on the $1$-skeleton of the associahedron is $O(n^3\log^3 n)$. We obtain similar rapid mixing results for the simple random walks on the $1$-skeleta of the type-$B$ and type-$D$ associahedra. We adapt Eppstein and Frishberg's technique to obtain the same bound of $O(n^3\log^3 n)$ in type $B$ and a bound of $O(n^{13} \log^2 n)$ in type $D$; in the process, we establish an expansion bound that is tight up to logarithmic factors in type $B$.

math.CO↗

arXiv2024

Random Subwords and Pipe Dreams

Fix a probability $p\in(0,1)$. Let $s_i$ denote the transposition in the symmetric group $\mathfrak{S}_n$ that swaps $i$ and $i+1$. Given a word $\mathsf{w}$ over the alphabet $\{s_1,\ldots,s_{n-1}\}$, we can generate a random subword by independently deleting each letter of $\mathsf{w}$ with probability $1-p$. For a large class of starting words $\mathsf{w}$ -- including all alternating reduced words for the decreasing permutation -- we compute precise asymptotics (as $n\to\infty$) for the expected number of inversions of the permutation represented by the random subword. This result can also be seen as an asymptotic formula for the expected number of inversions of a permutation represented by a certain random (non-reduced) pipe dream. In the special case when $\mathsf{w}$ is the word $(s_{n-1})(s_{n-2}s_{n-1})\cdots(s_1s_2\cdots s_{n-1})$, we find that the expected number of inversions of the permutation represented by the random subword is asymptotically equal to \[\frac{2\sqrt{2}}{3\sqrtπ}\sqrt{\frac{p}{1-p}}\,n^{3/2};\] this settles a conjecture of Morales, Panova, Petrov, and Yeliussizov.

math.PR↗

arXiv2024

Foot-Sorting for Socks

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 stack-sorting algorithm for set partitions. The sock orderings that can be sorted using a fixed number of feet are characterized by Klazar's notion of set partition pattern containment. We give an enumeration involving Fibonacci numbers for the $1$-foot-sortable sock orderings within a naturally-arising class. We also prove that if you have socks of $n$ different colors, then you can always sort them using at most $\left\lceil\log_2(n)\right\rceil$ feet, and we use a Ramsey-theoretic argument to show that this bound is tight.

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arXiv2024

Rainbow Stackings of Random Edge-Colorings

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 threshold for $r$ (as a function of $m$ and $n$) governing the existence and nonexistence of rainbow stackings of random $r$-edge-colorings $χ_1,\ldots,χ_m$.

math.CO↗

arXiv2024

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

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 representation-theoretic interpretations of a number of noteworthy families of type-$B$ quasisymmetric functions. Next, we construct modules of the type-$B$ $0$-Hecke algebra corresponding to type-$B$ analogues of Schur functions and introduce a type-$B$ analogue of Schur $Q$-functions; we prove that these shifted domino functions expand positively in the type-$B$ peak functions. We define a type-$B$ analogue of the $0$-Hecke--Clifford algebra, and we use this to provide representation-theoretic interpretations for both the type-$B$ peak functions and the shifted domino functions. We consider the modules of this algebra induced from type-$B$ $0$-Hecke modules constructed via ascent-compatibility and prove a general formula, in terms of type-$B$ peak functions, for the type-$B$ quasisymmetric characteristics of the restrictions of these modules.

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arXiv2024

Tilings of Benzels via Generalized Compression

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 allowed orientations of the tiles). Their primary tool, a bijection called compression that converts certain $k$-ribbon tilings to $(k-1)$-ribbon tilings, allowed them to reduce their problems to the enumeration of dimers (i.e., perfect matchings) of certain graphs. We present a generalized version of compression that no longer relies on the perspective of partitions and skew shapes. Using this strengthened tool, we resolve three more of Propp's conjectures and recast several others as problems about perfect matchings.

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arXiv2024

Rowmotion on $m$-Tamari and BiCambrian Lattices

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 when $b\equiv 1\pmod a$; in fact, we determine the entire orbit structure of rowmotion in this case, showing that it exhibits the cyclic sieving phenomenon. We additionally show that the down-degree statistic is homomesic for this action. In a different vein, we consider the action of rowmotion on Barnard and Reading's biCambrian lattices. Settling a different conjecture of Thomas and Williams, we prove that if $c$ is a bipartite Coxeter element of a coincidental-type Coxeter group $W$, then the orbit structure of rowmotion on the $c$-biCambrian lattice is the same as the orbit structure of rowmotion on the lattice of order ideals of the doubled root poset of type $W$.

math.CO↗

arXiv2024

Operahedron Lattices

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 resulting poset is a lattice. We answer this question in the affirmative, motivating us to name Laplante-Anfossi's posets operahedron lattices. The operahedron lattice of a chain with $n+1$ vertices is isomorphic to the $n$-th Tamari lattice, while the operahedron lattice of a claw with $n+1$ vertices is isomorphic to $\mathrm{Weak}(\mathfrak S_n)$, the weak order on the symmetric group $\mathfrak S_n$. We characterize semidistributive operahedron lattices and trim operahedron lattices. Let $Δ_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ be the principal order ideal of $\mathrm{Weak}(\mathfrak S_n)$ generated by the permutation ${w_\circ(k,n)=k(k-1)\cdots 1(k+1)(k+2)\cdots n}$. Our final result states that the operahedron lattice of a broom with $n+1$ vertices and $k$ leaves is isomorphic to the subposet of $\mathrm{Weak}(\mathfrak S_n)$ consisting of the preimages of $Δ_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ under West's stack-sorting map; as a consequence, we deduce that this subposet is a semidistributive lattice.

math.CO↗

arXiv2024

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

Reading cut the hyperplanes in a real central arrangement $\mathcal H$ into pieces called \emph{shards}, which reflect order-theoretic properties of the arrangement. We show that shards have a natural interpretation as certain generators of the fundamental group of the complement of the complexification of $\mathcal H$. Taking only positive expressions in these generators yields a new poset that we call the \emph{pure shard monoid}. When $\mathcal H$ is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator $\mathsf{Pop}$. In this case, we use $\mathsf{Pop}$ to define an embedding $\mathsf{Crackle}$ of Reading's shard intersection order into the pure shard monoid. When $\mathcal H$ is the reflection arrangement of a finite Coxeter group, we also define a poset embedding $\mathsf{Snap}$ of the shard intersection order into the positive braid monoid; in this case, our three maps are related by $\mathsf{Snap}=\mathsf{Crackle} \cdot \mathsf{Pop}$.

math.CO↗

arXiv2024

The Ungar Games

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 players -- Atniss and Eeta -- take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say $L$ is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on $L$. We first prove that the number of principal order ideals in the weak order on $S_n$ that are Eeta wins is $O(0.95586^nn!)$. We then consider a broad class of intervals in Young's lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type-$A$ root posets. We also characterize and enumerate principal order ideals in Tamari lattices that are Eeta wins. Finally, we conclude with some open problems and a short discussion of the computational complexity of Ungar games.

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arXiv2023

Pop-Stack Operators for Torsion Classes and Cambrian Lattices

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 $x$ in $L$. We study several properties of the pop-stack operator of $\mathrm{tors}Λ$, the lattice of torsion classes of a $τ$-tilting finite algebra $Λ$ over a field $K$. We describe the pop-stack operator in terms of certain mutations of 2-term simple-minded collections. This allows us to describe preimages of a given torsion class under the pop-stack operator. We then specialize our attention to Cambrian lattices of a finite irreducible Coxeter group $W$. Using tools from representation theory, we provide simple Coxeter-theoretic and lattice-theoretic descriptions of the image of the pop-stack operator of a Cambrian lattice (which can be stated without representation theory). When specialized to a bipartite Cambrian lattice of type A, this result settles a conjecture of Choi and Sun. We also settle a related enumerative conjecture of Defant and Williams. When $L$ is an arbitrary lattice quotient of the weak order on $W$, we prove that the maximum size of a forward orbit under the pop-stack operator of $L$ is at most the Coxeter number of $W$; when $L$ is a Cambrian lattice, we provide an explicit construction to show that this maximum forward orbit size is actually equal to the Coxeter number.

math.CO↗

arXiv2023

Rowmotion Markov Chains

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 ${\bf M}_{J(P)}$ by assigning a probability $p_x$ to each $x\in P$ and using these probabilities to insert randomness into the original definition of rowmotion. More generally, we introduce a very broad family of toggle Markov chains inspired by Striker's notion of generalized toggling. We characterize when toggle Markov chains are irreducible, and we show that each toggle Markov chain has a remarkably simple stationary distribution. We also provide a second generalization of rowmotion Markov chains to the context of semidistrim lattices. Given a semidistrim lattice $L$, we assign a probability $p_j$ to each join-irreducible element $j$ of $L$ and use these probabilities to construct a rowmotion Markov chain ${\bf M}_L$. Under the assumption that each probability $p_j$ is strictly between $0$ and $1$, we prove that ${\bf M}_{L}$ is irreducible. We also compute the stationary distribution of the rowmotion Markov chain of a lattice obtained by adding a minimal element and a maximal element to a disjoint union of two chains. We bound the mixing time of ${\bf M}_{L}$ for an arbitrary semidistrim lattice $L$. In the special case when $L$ is a Boolean lattice, we use spectral methods to obtain much stronger estimates on the mixing time, showing that rowmotion Markov chains of Boolean lattices exhibit the cutoff phenomenon.

math.CO↗

arXiv2023

Ordering Candidates via Vantage Points

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 the points of $V$. Let $Ψ_k(C)$ denote the set of orderings of $C$ that can be obtained in this manner as $V$ varies, and let $ψ^{\mathrm{max}}_{d,k}(n)$ be the maximum of $\lvertΨ_k(C)\rvert$ as $C$ ranges over all $n$-element subsets of $\mathbb{R}^d$. We prove that $ψ^{\mathrm{max}}_{d,k}(n)=Θ_{d,k}(n^{2dk})$ when $d \geq 2$ and that $ψ^{\mathrm{max}}_{1,k}(n)=Θ_k(n^{4\lceil k/2\rceil -1})$. As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set $Ψ(C)=\bigcup_{k\geq 1}Ψ_k(C)$; this includes an exact description of $Ψ(C)$ when $d=1$ and when $C$ is the set of vertices of a vertex-transitive polytope.

math.CO↗

arXiv2023

Homomesy via Toggleability Statistics

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 every orbit of the action. We systematize a technique for proving that various statistics of interest are homomesic by writing these statistics as linear combinations of "toggleability statistics" (originally introduced by Striker) plus a constant. We show that this technique recaptures most of the known homomesies for the posets on which rowmotion has been most studied. We also show that the technique continues to work in modified contexts. For instance, this technique also yields homomesies for the piecewise-linear and birational extensions of rowmotion; furthermore, we introduce a $q$-analogue of rowmotion and show that the technique yields homomesies for "$q$-rowmotion" as well.

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