arXiv subjects

Colin Defant

Publications and source records attributed to Colin Defant.

At least 55 records · Page 3Linked to original sources

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.

math.CO↗

arXiv2023

Extensions of Hitomezashi Patterns

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 context of percolation theory, and the first two authors recently investigated additional structural properties of them. In this paper, we establish several optimization-style results on hitomezashi patterns and provide a complete classification of "long-stitch" hitomezashi patterns in which stitches have length greater than 1. We also study variants in which stitches can have directions not parallel to the coordinate axes.

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arXiv2023

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

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 toggle operators, listed in a natural cyclic order. We consider more general permutoric promotion operators, which are defined as compositions of the same toggles, but in permuted orders. We settle a conjecture of the first author by determining the orders of all permutoric promotion operators when $G$ is a path graph. In fact, we completely characterize the orbit structures of these operators, showing that they satisfy the cyclic sieving phenomenon. The first half of our proof requires us to introduce and analyze new broken promotion operators, which can be interpreted via globs of liquid gliding on a path graph. For the latter half of our proof, we reformulate the dynamics of permutoric promotion via stones sliding along a cycle graph and coins colliding with each other on a path graph.

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arXiv2023

Fertilitopes

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West's stack-sorting map $s$. Associated to each permutation $π$ is a particular set $\mathcal V(π)$ of integer compositions that appears in a formula for the fertility of $π$, which is defined to be $|s^{-1}(π)|$. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that $\mathcal V(π)$ is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of $π$ to be the convex hull of $\mathcal V(π)$, and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of $π$ directly from $π$ using Bousquet-Mélou's notion of the canonical tree of $π$. As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density $1$, and we determine all infertility numbers of size at most $126$. Finally, we reformulate the conjecture that $\sum_{σ\in s^{-1}(π)}x^{\text{des}(σ)+1}$ is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

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arXiv2023

Torsors and tilings from toric toggling

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 path graph. We investigate a "toric" analogue of this work by analyzing the dynamics arising from toggling independent sets of a cycle graph. Each orbit in the dynamical system can be encoded via a grid of 0s and 1s; two commuting bijections on the set of 1s in this grid produce torsors for what we call the infinite snake group and the finite ouroboros groups. By studying related covering maps, we deduce precise combinatorial properties of the orbits. Because the snake and ouroboros groups are abelian, they define tilings of cylinders and tori by parallelograms, which we also characterize. Many of the ideas developed here should be adaptable both to other toggle actions in combinatorics and to other cellular automata.

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arXiv2023

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 Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.

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arXiv2023

Motzkin Intervals and Valid Hook Configurations

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 configurations of $312$-avoiding permutations and intervals in these new posets. We also show that valid hook configurations of permutations avoiding $132$ (or equivalently, $231$) are counted by the same numbers that count intervals in the Motzkin-Tamari posets that Fang recently introduced, and we give an asymptotic formula for these numbers. We then proceed to enumerate valid hook configurations of permutations avoiding other collections of patterns. We also provide enumerative conjectures, one of which links valid hook configurations of $312$-avoiding permutations, intervals in the new posets we have defined, and certain closed lattice walks with small steps that are confined to a quarter plane.

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arXiv2023

Triangular-Grid Billiards and Plabic Graphs

Given a polygon $P$ in the triangular grid, we obtain a permutation $π_P$ via a natural billiards system in which beams of light bounce around inside of $P$. The different cycles in $π_P$ correspond to the different trajectories of light beams. We prove that \[\text{area}(P)\geq 6\text{cyc}(P)-6\quad\text{and}\quad\text{perim}(P)\geq\frac{7}{2}\text{cyc}(P)-\frac{3}{2},\] where $\text{area}(P)$ and $\text{perim}(P)$ are the (appropriately normalized) area and perimeter of $P$, respectively, and $\text{cyc}(P)$ is the number of cycles in $π_P$. The inequality concerning $\text{area}(P)$ is tight, and we characterize the polygons $P$ satisfying $\text{area}(P)=6\text{cyc}(P)-6$. These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let $G$ be a connected reduced plabic graph with essential dimension $2$. Suppose $G$ has $n$ marked boundary points and $v$ (internal) vertices, and let $c$ be the number of cycles in the trip permutation of $G$. Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]

math.CO↗

arXiv2023

Ungarian Markov Chains

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 $T$ is a random subset of the set of elements covered by $x$. We focus on estimating $\mathcal E(L)$, the expected number of steps of ${\bf U}_L$ needed to get from the top element of $L$ to the bottom element of $L$. Using direct combinatorial arguments, we provide asymptotic estimates when $L$ is the weak order on the symmetric group $S_n$ and when $L$ is the $n$-th Tamari lattice. When $L$ is distributive, the Markov chain ${\bf U}_L$ is equivalent to an instance of the well-studied random process known as last-passage percolation with geometric weights. One of our main results states that if $L$ is a trim lattice, then $\mathcal E(L)\leq\mathcal E(\text{spine}(L))$, where $\text{spine}(L)$ is a specific distributive sublattice of $L$ called the spine of $L$. Combining this lattice-theoretic theorem with known results about last-passage percolation yields a powerful method for proving upper bounds for $\mathcal E(L)$ when $L$ is trim. We apply this method to obtain uniform asymptotic upper bounds for the expected number of steps in the Ungarian Markov chains of Cambrian lattices of classical types and the Ungarian Markov chains of $ν$-Tamari lattices.

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arXiv2022

Tilings of Benzels via the Abacus Bijection

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 prove some additional results that he left tacit. In order to solve these problems, we first transfer benzels into the square grid. One of our primary tools, which we combine with several new ideas, is a bijection (rediscovered by Stanton and White and often attributed to them although it is considerably older) between $k$-ribbon tableaux of certain skew shapes and certain $k$-tuples of Young tableaux.

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arXiv2022

Toric Promotion

We introduce toric promotion as a cyclic analogue of Schützenberger's promotion operator. Toric promotion acts on the set of labelings of a graph $G$. We discuss connections between toric promotion and previously-studied notions such as toric posets and friends-and-strangers graphs. Our main theorem provides a surprisingly simple description of the orbit structure of toric promotion when $G$ is a forest.

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arXiv2022

Connectedness and Cycle Spaces of Friends-and-Strangers Graphs

If $X=(V(X),E(X))$ and $Y=(V(Y),E(Y))$ are $n$-vertex graphs, then their friends-and-strangers graph $\mathsf{FS}(X,Y)$ 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 if there is an edge $\{a,b\}\in E(X)$ such that $\{σ(a),σ(b)\}\in E(Y)$ and $σ'=σ\circ (a\,\,b)$, where $(a\,\,b)$ is the permutation of $V(X)$ that swaps $a$ and $b$. We prove general theorems that provide necessary and/or sufficient conditions for $\mathsf{FS}(X,Y)$ to be connected. As a corollary, we obtain a complete characterization of the graphs $Y$ such that $\mathsf{FS}(\mathsf{Dand}_{k,n},Y)$ is connected, where $\mathsf{Dand}_{k,n}$ is a dandelion graph; this substantially generalizes a theorem of the first author and Kravitz in the case $k=3$. For specific choices of $Y$, we characterize the spider graphs $X$ such that $\mathsf{FS}(X,Y)$ is connected. In a different vein, we study the cycle spaces of friends-and-strangers graphs. Naatz proved that if $X$ is a path graph, then the cycle space of $\mathsf{FS}(X,Y)$ is spanned by $4$-cycles and $6$-cycles; we show that the same statement holds when $X$ is a cycle and $Y$ has domination number at least $3$. When $X$ is a cycle and $Y$ has domination number at least $2$, our proof sheds light on how walks in $\mathsf{FS}(X,Y)$ behave under certain Coxeter moves.

math.CO↗

arXiv2022

Pop-Stack-Sorting for Coxeter Groups

Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $\mathsf{Pop}:W\to W$ to be the map that fixes the identity element and sends each nonidentity element $w$ to the meet of the elements covered by $w$ in the right weak order. When $W$ is the symmetric group $S_n$, $\mathsf{Pop}$ coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where $h$ is the Coxeter number of $W$ (with $h=\infty$ if $W$ is infinite) and $O_f(w)$ denotes the forward orbit of $w$ under a map $f$. When $W$ is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of $W$ is $h-1$. More generally, we define a map $f:W\to W$ to be compulsive if for every $w\in W$, $f(w)$ is less than or equal to $\mathsf{Pop}(w)$ in the right weak order. We prove that if $f$ is compulsive, then $\sup\limits_{w\in W}|O_f(w)|\leq h$. This result is new even for symmetric groups. We prove that $2$-pop-stack-sortable elements in type $B$ are in bijection with $2$-pop-stack-sortable permutations in type $A$, which were enumerated by Pudwell and Smith. Claesson and Gudmundsson proved that for each fixed nonnegative integer $t$, the generating function that counts $t$-pop-stack-sortable permutations in type $A$ is rational; we establish analogous results in types $B$ and $\widetilde A$.

math.CO↗

arXiv2022

Loops and Regions in Hitomezashi Patterns

Hitomezashi patterns, which originate from traditional Japanese embroidery, are intricate arrangements of unit-length line segments called stitches. The stitches connect to form hitomezashi strands and hitomezashi loops, which divide the plane into regions. We investigate the deeper mathematical properties of these patterns, which also feature prominently in the study of corner percolation. It was previously known that every loop in a hitomezashi pattern has odd width and odd height. We additionally prove that such a loop has length congruent to $4$ modulo $8$ and area congruent to $1$ modulo $4$. Although these results are simple to state, their proofs require us to understand the delicate topological and combinatorial properties of slicing operations that can be applied to hitomezashi patterns. We also show that the expected number of regions in a random $m\times n$ hitomezashi pattern (chosen according to a natural random model) is asymptotically $\left(\frac{π^2-9}{12}+o(1)\right)mn$.

math.CO↗

arXiv2022

Stack-Sorting for Coxeter Groups

Given an essential semilattice congruence $\equiv$ on the left weak order of a Coxeter group $W$, we define the Coxeter stack-sorting operator ${\bf S}_\equiv:W\to W$ by ${\bf S}_\equiv(w)=w\left(π_\downarrow^\equiv(w)\right)^{-1}$, where $π_\downarrow^\equiv(w)$ is the unique minimal element of the congruence class of $\equiv$ containing $w$. When $\equiv$ is the sylvester congruence on the symmetric group $S_n$, the operator ${\bf S}_\equiv$ is West's stack-sorting map. When $\equiv$ is the descent congruence on $S_n$, the operator ${\bf S}_\equiv$ is the pop-stack-sorting map. We establish several general results about Coxeter stack-sorting operators, especially those acting on symmetric groups. For example, we prove that if $\equiv$ is an essential lattice congruence on $S_n$, then every permutation in the image of ${\bf S}_\equiv$ has at most $\left\lfloor\frac{2(n-1)}{3}\right\rfloor$ right descents; we also show that this bound is tight. We then introduce analogues of permutree congruences in types $B$ and $\widetilde A$ and use them to isolate Coxeter stack-sorting operators $\mathtt{s}_B$ and $\widetilde{\hspace{.05cm}\mathtt{s}}$ that serve as canonical type-$B$ and type-$\widetilde A$ counterparts of West's stack-sorting map. We prove analogues of many known results about West's stack-sorting map for the new operators $\mathtt{s}_B$ and $\widetilde{\hspace{.05cm}\mathtt{s}}$. For example, in type $\widetilde A$, we obtain an analogue of Zeilberger's classical formula for the number of $2$-stack-sortable permutations in $S_n$.

math.CO↗

arXiv2022

Troupes, Cumulants, and Stack-Sorting

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 operations on colored binary plane trees that we call insertion and decomposition, we prove that this surprising phenomenon holds for families of trees that we call troupes. We give a simple characterization of troupes, which provide a broad framework for generalizing several of the results known about West's stack-sorting map $s$. Indeed, we give new proofs of some of the main techniques that have been developed for understanding $s$; these new proofs are far more conceptual than the original ones, explain how the objects called valid hook configurations arise naturally, and generalize to troupes. For $t\in\{2,3\}$, we enumerate $t$-stack-sortable alternating permutations of odd length and $t$-stack-sortable permutations whose descents are all peaks. The unexpected connection between troupes and cumulants provides a powerful new tool for analyzing the stack-sorting map that hinges on free probability theory. We give numerous applications of this method. For example, we show that if $σ\in S_{n-1}$ is chosen uniformly at random, then the expected value of $\text{des}(s(σ))+1$ is \[\left(3-\sum_{j=0}^n\frac{1}{j!}\right)n.\] Furthermore, the variance of $\text{des}(s(σ))+1$ is asymptotically $(2+2e-e^2)n$. We obtain similar results concerning the expected number of descents of postorder readings of decreasing colored binary plane trees. We also obtain improved estimates for $|s(S_n)|$ and an improved lower bound for the degree of noninvertibility of $s$. We give two novel formulas that convert from free to classical cumulants. The first is given by a sum over noncrossing partitions, and the second is given by a sum over $231$-avoiding valid hook configurations. We pose several open problems.

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arXiv2021

Meeting Covered Elements in $ν$-Tamari Lattices

For each complete meet-semilattice $M$, we define an operator $\mathsf{Pop}_M:M\to M$ by \[\mathsf{Pop}_M(x)=\bigwedge(\{y\in M:y\lessdot x\}\cup\{x\}).\] When $M$ is the right weak order on a symmetric group, $\mathsf{Pop}_M$ is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of $\mathsf{Pop}_{\text{Tam}(ν)}$, where $\text{Tam}(ν)$ is the $ν$-Tamari lattice. We determine the maximum size of a forward orbit of $\mathsf{Pop}_{\text{Tam}(ν)}$. When $\text{Tam}(ν)$ is the $n^\text{th}$ $m$-Tamari lattice, this maximum forward orbit size is $m+n-1$; in this case, we prove that the number of forward orbits of size $m+n-1$ is \[\frac{1}{n-1}\binom{(m+1)(n-2)+m-1}{n-2}.\] Motivated by the recent investigation of the pop-stack-sorting map, we define a lattice path $μ\in\text{Tam}(ν)$ to be $t$-$\mathsf{Pop}$-sortable if $\mathsf{Pop}_{\text{Tam}(ν)}^t(μ)=ν$. We enumerate $1$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(ν)$ for arbitrary $ν$. We also give a recursive method to generate $2$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(ν)$ for arbitrary $ν$; this allows us to enumerate $2$-$\mathsf{Pop}$-sortable lattice paths in a large variety of $ν$-Tamari lattices that includes the $m$-Tamari lattices.

math.CO↗

arXiv2021

Semidistrim Lattices

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 in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim, and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere. Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard's $\overlineκ$ map on semidistributive lattices as well as Thomas and the second author's rowmotion on trim lattices. Every lattice has an associated pop-stack sorting operator that sends an element $x$ to the meet of the elements covered by $x$. For semidistrim lattices, we are able to derive several intimate connections between rowmotion and pop-stack sorting, one of which involves independent dominating sets of the Galois graph.

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