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Colin Defant

Publications and source records attributed to Colin Defant.

At least 73 records · Page 4Linked to original sources

arXiv2021

Crystal Pop-Stack Sorting and Type A Crystal Lattices

Given a complex simple Lie algebra $\mathfrak g$ and a dominant weight $λ$, let $\mathcal B_λ$ be the crystal poset associated to the irreducible representation of $\mathfrak g$ with highest weight $λ$. In the first part of the article, we introduce the \emph{crystal pop-stack sorting operator} $\mathsf{Pop}_{\lozenge}\colon\mathcal B_λ\to\mathcal B_λ$, a noninvertible operator whose definition extends that of the pop-stack sorting map and the recently-introduced Coxeter pop-stack sorting operators. Every forward orbit of $\mathsf{Pop}_{\lozenge}$ contains the minimal element of $\mathcal B_λ$, which is fixed by $\mathsf{Pop}_{\lozenge}$. We prove that the maximum size of a forward orbit of $\mathsf{Pop}_{\lozenge}$ is the Coxeter number of the Weyl group of $\mathfrak g$. In the second part of the article, we characterize exactly when a type $A$ crystal is a lattice.

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arXiv2021

Symmetry of Narayana numbers and rowvacuation of root posets

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 keeping track of the cardinalities of these antichains. The $W$-Narayana numbers are symmetric, i.e., the number of antichains of cardinality $k$ is the same as the number of cardinality $r-k$. However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the $W$-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of "toggles," that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev's desired involution.

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arXiv2021

The runsort permuton

Suppose we choose a permutation $π$ uniformly at random from $S_n$. Let $\mathsf{runsort}(π)$ be the permutation obtained by sorting the ascending runs of $π$ into lexicographic order. Alexandersson and Nabawanda recently asked if the plot of $\mathsf{runsort}(π)$, when scaled to the unit square $[0,1]^2$, converges to a limit shape as $n\to\infty$. We answer their question by showing that the measures corresponding to the scaled plots of these permutations $\mathsf{runsort}(π)$ converge with probability $1$ to a permuton (limiting probability distribution) that we describe explicitly. In particular, the support of this permuton is $\{(x,y)\in[0,1]^2:x\leq ye^{1-y}\}$.

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arXiv2021

Friends and Strangers Walking on Graphs

Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, we define a graph $\mathsf{FS}(X,Y)$ whose vertex set consists of all bijections $σ:V(X)\to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if they agree everywhere except for two adjacent vertices $a,b \in V(X)$ such that $σ(a)$ and $σ(b)$ are adjacent in $Y$. This setup, which has a natural interpretation in terms of friends and strangers walking on graphs, provides a common generalization of Cayley graphs of symmetric groups generated by transpositions, the famous $15$-puzzle, generalizations of the $15$-puzzle as studied by Wilson, and work of Stanley related to flag $h$-vectors. We derive several general results about the graphs $\mathsf{FS}(X,Y)$ before focusing our attention on some specific choices of $X$. When $X$ is a path graph, we show that the connected components of $\mathsf{FS}(X,Y)$ correspond to the acyclic orientations of the complement of $Y$. When $X$ is a cycle, we obtain a full description of the connected components of $\mathsf{FS}(X,Y)$ in terms of toric acyclic orientations of the complement of $Y$. We then derive various necessary and/or sufficient conditions on the graphs $X$ and $Y$ that guarantee the connectedness of $\mathsf{FS}(X,Y)$. Finally, we raise several promising further questions.

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arXiv2021

Typical and Extremal Aspects of Friends-and-Strangers Graphs

Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $σ:V(X)\to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if they agree everywhere except for two adjacent vertices $a,b \in V(X)$ such that $σ(a)$ and $σ(b)$ are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected; we address this problem from two different perspectives. First, we address the case of "typical" $X$ and $Y$ by proving that if $X$ and $Y$ are independent Erdős-Rényi random graphs with $n$ vertices and edge probability $p$, then the threshold probability guaranteeing the connectedness of $\mathsf{FS}(X,Y)$ with high probability is $p=n^{-1/2+o(1)}$. Second, we address the case of "extremal" $X$ and $Y$ by proving that the smallest minimum degree of the $n$-vertex graphs $X$ and $Y$ that guarantees the connectedness of $\mathsf{FS}(X,Y)$ is between $3n/5+O(1)$ and $9n/14+O(1)$. When $X$ and $Y$ are bipartite, a parity obstruction forces $\mathsf{FS}(X,Y)$ to be disconnected. In this bipartite setting, we prove analogous "typical" and "extremal" results concerning when $\mathsf{FS}(X,Y)$ has exactly $2$ connected components; for the extremal question, we obtain a nearly exact result.

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arXiv2021

Coxeter Pop-Tsack Torsing

Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$, we define the Coxeter pop-tsack torsing operator $\mathsf{Pop}_T:W\to W$ by $\mathsf{Pop}_T(w)=w\cdotπ_T(w)^{-1}$, where $π_T(w)$ is the join in the noncrossing partition lattice $\mathrm{NC}(w,c)$ of the set of reflections lying weakly below $w$ in the absolute order. This definition serves as a "Bessis dual" version of the first author's notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack-sorting map on symmetric groups. We show that if $W$ is coincidental or of type $D$, then the identity element of $W$ is the unique periodic point of $\mathsf{Pop}_T$ and the maximum size of a forward orbit of $\mathsf{Pop}_T$ is the Coxeter number $h$ of $W$. In each of these types, we obtain a natural lift from $W$ to the dual braid monoid of $W$. We also prove that $W$ is coincidental if and only if it has a unique forward orbit of size $h$. For arbitrary $W$, we show that the forward orbit of $c^{-1}$ under $\mathsf{Pop}_T$ has size $h$ and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.

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arXiv2021

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

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 first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by ${\bf B}\circ s$, where ${\bf B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(τ^{(1)},\ldots,τ^{(r)}))$ for $\{τ^{(1)},\ldots,τ^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, where $\text{Av}_{n,k}(231,312,321)$ is the set of permutations in $\text{Av}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice.

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arXiv2020

Highly Sorted Permutations and Bell Numbers

Let $s$ denote West's stack-sorting map. For all positive integers $m$ and all integers $n\geq 2m-2$, we give a simple characterization of the set $s^{n-m}(S_n)$; as a consequence, we find that $|s^{n-m}(S_n)|$ is the $m^\text{th}$ Bell number $B_m$. We also prove that the restriction $n\geq 2m-2$ is tight by showing that $|s^{m-3}(S_{2m-3})|=B_m+m-2$ for all $m\geq 3$.

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arXiv2020

Asymptotics of 3-stack-sortable permutations

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we conjecture that the generating function behaves as $$W(t) \sim C_0(1-μ_3 t)^α\cdot \log^β(1-μ_3 t), $$ so that $$[t^n]W(t)=w_n \sim \frac{c_0μ_3^n}{ n^{(α+1)}\cdot \log^λ{n}} ,$$ where $μ_3 = 9.69963634535(30),$ $α= 2.0 \pm 0.25.$ If $α= 2$ exactly, then $λ= -β+1$, and we estimate $β\approx -3.$ If $α$ is not an integer, then $λ=-β$, but we cannot give a useful estimate of $β$. The growth constant estimate (just) contradicts a conjecture of the first author that $$9.702 < μ_3 \le 9.704.$$ We also prove a new rigorous lower bound of $μ_3\geq 9.4854$, allowing us to disprove a conjecture of Bóna. We then further extend the series using differential-approximants to obtain approximate coefficients $O(t^{2000}),$ expected to be accurate to $20$ significant digits, and use the approximate coefficients to provide additional evidence supporting the results obtained from the exact coefficients.

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arXiv2020

Polyurethane Toggles

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 involutions so that they generate a group $\mathscr P_n$ that acts on the set $S_n$ of permutations of $\{1,\ldots,n\}$. After characterizing its orbits in terms of permutation skeletons, we apply the action in order to understand West's stack-sorting map. We obtain a very simple proof of a result that clarifies and extensively generalizes a theorem of Bouvel and Guibert and also generalizes a theorem of Bousquet-Mélou. We also settle a conjecture of Bouvel and Guibert. We prove a result related to the recently-introduced notion of postorder Wilf equivalence. Finally, we investigate an interesting connection among the action of $\mathscr P_n$ on $S_n$, the group structure of $S_n$, and the stack-sorting map.

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arXiv2020

Quantifying Noninvertibility in Discrete Dynamical Systems

Given a finite set $X$ and a function $f:X\to X$, we define the degree of noninvertibility of $f$ to be $\displaystyle\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\text{deg}(f^k)}{\text{deg}(f)^γ}=Θ(n^{1-1/2^{k-1}})\] for every real number $γ\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.

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arXiv2020

Fertility Monotonicity and Average Complexity of the Stack-Sorting Map

Let $\mathcal D_n$ 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 $123\cdots n$. We prove that \[0.62433\approxλ\leq\liminf_{n\to\infty}\frac{\mathcal D_n}{n}\leq\limsup_{n\to\infty}\frac{\mathcal D_n}{n}\leq \frac{3}{5}(7-8\log 2)\approx 0.87289,\] where $λ$ is the Golomb-Dickman constant. Our lower bound improves upon West's lower bound of $0.23$, and our upper bound is the first improvement upon the trivial upper bound of $1$. We then show that fertilities of permutations increase monotonically upon iterations of $s$. More precisely, we prove that $|s^{-1}(σ)|\leq|s^{-1}(s(σ))|$ for all $σ\in S_n$, where equality holds if and only if $σ=123\cdots n$. This is the first theorem that manifests a law-of-diminishing-returns philosophy for the stack-sorting map that Bóna has proposed. Along the way, we note some connections between the stack-sorting map and the right and left weak orders on $S_n$.

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arXiv2020

Stack-Sorting with Consecutive-Pattern-Avoiding Stacks

We introduce consecutive-pattern-avoiding stack-sorting maps $\text{SC}_σ$, which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_σ$ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns $σ$ such that $\text{Sort}(\text{SC}_σ)$, the set of permutations that are sortable via the map $s\circ\text{SC}_σ$, is a permutation class, and we enumerate the sets $\text{Sort}(\text{SC}_σ)$ for $σ\in\{123,132,321\}$. We also study the maps $\text{SC}_σ$ from a dynamical point of view, characterizing the periodic points of $\text{SC}_σ$ for all $σ\in S_3$ and computing $\max_{π\in S_n}|\text{SC}_σ^{-1}(π)|$ for all $σ\in\{132,213,231,312\}$. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map $s_{132}$, and we show that the maximum number of iterations of $s_{132}$ needed to send a permutation in $S_n$ to a periodic point is $n-1$. The paper ends with numerous open problems and conjectures.

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arXiv2020

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

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting map. We prove that the sequences obtained by counting these permutations according to their first entries are symmetric, and we conjecture that they are log-concave. We also obtain new recurrence relations involving Lassalle's sequence and the sequence that enumerates valid hook configurations. We end with several suggestions for future work.

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arXiv2020

Pattern-Avoiding Permutation Powers

Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation $π$ strongly avoids a pattern $τ$ if $π$ and $π^2$ both avoid $τ$. They conjectured that for every positive integer $k$, there is a permutation in $S_{k^3}$ that strongly avoids $123\cdots (k+1)$. We use the Robinson--Schensted--Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least $k^{k^3/2+O(k^3/\log k)}$ and at most $k^{2k^3+O(k^3/\log k)}$. We enumerate $231$-avoiding permutations of order $3$, and we give two further enumerative results concerning strong pattern avoidance. We also consider permutations whose powers all avoid a pattern $τ$. Finally, we study subgroups of symmetric groups whose elements all avoid certain patterns. This leads to several new open problems connecting the group structures of symmetric groups with pattern avoidance.

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arXiv2020

Supertrees

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 $k$-superpermutation has recently received significant attention in the field of permutation patterns. One can ask analogous questions for other classes of objects. In this paper, we study $k$-supertrees. For each $d\geq 2$, we focus on two types of rooted plane trees called $d$-ary plane trees and $[d]$-trees. Motivated by recent developments in the literature, we consider "contiguous" and "noncontiguous" notions of pattern containment for each type of tree. We obtain both upper and lower bounds on the minimum possible size of a $k$-supertree in three cases; in the fourth, we determine the minimum size exactly. One of our lower bounds makes use of a recent result of Albert, Engen, Pantone, and Vatter on $k$-universal layered permutations.

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arXiv2020

Promotion Sorting

Schützenberger'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 operator that acts on all labelings of a poset. We prove several properties of $\partial$; in particular, we show that for every labeling $L$ of an $n$-element poset $P$, the labeling $\partial^{n-1}(L)$ is a linear extension of $P$. Thus, we can view the dynamical system defined by $\partial$ as a sorting procedure that sorts labelings into linear extensions. For all $0\leq k\leq n-1$, we characterize the $n$-element posets $P$ that admit labelings that require at least $n-k-1$ iterations of $\partial$ in order to become linear extensions. The case in which $k=0$ concerns labelings that require the maximum possible number of iterations in order to be sorted; we call these labelings tangled. We explicitly enumerate tangled labelings for a large class of posets that we call inflated rooted forest posets. For an arbitrary finite poset, we show how to enumerate the sortable labelings, which are the labelings $L$ such that $\partial(L)$ is a linear extension.

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arXiv2020

Catalan Intervals and Uniquely Sorted Permutations

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 patterns are equinumerous with intervals in these posets. While most of our proofs are bijective, some use generating trees and generating functions. We end with several conjectures.

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