arXiv subjects

Noah Kravitz

Publications and source records attributed to Noah Kravitz.

At least 37 records · Page 2Linked to original sources

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 $\widetilde G_2$.

math.CO↗

arXiv2024

Relative Lonely Runner spectra

For a subtorus $T \subseteq (\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the $L^\infty$-distance from $T$ to the point $(1/2, \ldots, 1/2)$. For a subtorus $U \subseteq (\mathbb{R}/\mathbb{Z})^n$, define $\mathcal{S}_1(U)$, the Lonely Runner spectrum relative to $U$, to be the set of all values of $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $U$ not contained in the union of the coordinate hyperplanes of $(\mathbb{R}/\mathbb{Z})^n$. The relative spectrum $\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)$ is the ordinary Lonely Runner spectrum that has been studied previously. Giri and the second author recently showed that the relative spectra $\mathcal{S}_1(U)$ for $2$-dimensional subtori $U \subseteq (\mathbb{R}/\mathbb{Z})^n$ essentially govern the accumulation points of the Lonely Runner spectrum $\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)$. In the present work, we prove that such relative spectra $\mathcal{S}_1(U)$ have a very rigid arithmetic structure, and that one can explicitly find a complete characterization of each such relative spectrum with a finite calculation; carrying out this calculation for a few specific examples sheds light on previous constructions in the literature on the Lonely Runner Problem.

math.NT↗

arXiv2024

Reconstructing a set from its subset sums: $2$-torsion-free groups

For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for $2$-torsion-free groups $G$ by giving characterizations, both algebraic and combinatorial, of the fibers of $\text{FS}$. Equivalently, we characterize all pairs of multisets $A,B$ with $\text{FS}(A)=\text{FS}(B)$. Our results build on recent work of Ciprietti and the first author.

math.CO↗

arXiv2024

Corners with polynomial side length

A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In the case where $P$ has an integer root of multiplicity $1$, we show that if $A \subseteq [N]^2$ does not contain any nontrivial $P$-polynomial corners, then $$|A| \ll_P \frac{N^2}{(\log\log\log N)^c}$$ for some absolute constant $c>0$. This simultaneously generalizes a result of Shkredov about corner-free sets and a recent result of Peluse, Sah, and Sawhney about sets without $3$-term arithmetic progressions of common difference $z^2-1$. The main ingredients in our proof are a multidimensional quantitative concatenation result from our companion paper arXiv:2407.08636 and a novel degree-lowering argument for box norms.

math.CO↗

arXiv2024

Rearranging small sets for distinct partial sums

A conjecture of Graham (repeated by Erdős) asserts that for any set $A \subseteq \mathbb{F}_p \setminus \{0\}$, there is an ordering $a_1, \ldots, a_{|A|}$ of the elements of $A$ such that the partial sums $a_1, a_1+a_2, \ldots, a_1+a_2+\cdots+a_{|A|}$ are all distinct. We give a very short proof of this conjecture for sets $A$ of size at most $\log p/\log\log p$.

math.CO↗

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.

math.CO↗

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

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.

math.CO↗

arXiv2023

Hermite--Hadamard inequalities for nearly-spherical domains

A conjecture of Pasteczka, generalizing the classical Hermite--Hadamard Inequality, states that if $Ω\subseteq \mathbb{R}^d$ is a compact convex domain such that $Ω$ and $\partial Ω$ have the same center of mass, then for every convex function $f: Ω\to \mathbb{R}^d$, the average value of $f$ on $Ω$ is less than or equal to the average value of $f$ on $\partial Ω$. Pasteczka proved this conjecture for the case where $Ω$ is a polytope with an inscribed ball. We generalize this result by proving Pasteczka's conjecture in the case where some point lies at most $(d+1)|Ω|/|\partial Ω|$ away from all hyperplanes tangent to $\partial Ω$.

math.CA↗

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

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.

math.CO↗

arXiv2022

Counting Dope Matrices

For a polynomial $P$ of degree $n$ and an $m$-tuple $Λ=(λ_1,\dots,λ_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $Λ$ is $D_P(Λ)=(δ_{ij})_{i\in [1,m],j\in[0,n]}$, where $δ_{ij}=1$ if $P^{(j)}(λ_i)=0$, and $δ_{ij}=0$ otherwise. Our first result is a combinatorial characterization of the $2$-row dope matrices (for all pairs $Λ$); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of $m\times(n+1)$ dope matrices, and we show that the number of $m \times (n+1)$ dope matrices for a fixed $m$-tuple $Λ$ is maximized when $Λ$ is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems.

math.CO↗

arXiv2022

Cats in cubes

Answering a recent question of Patchell and Spiro, we show that when a $d$-dimensional cube of side length $n$ is filled with letters, the word $\mathsf{CAT}$ can appear contiguously at most $(3^{d-1}/2)n^d$ times (allowing diagonals); we also characterize when equality occurs and extend our results to words other than $\mathsf{CAT}$.

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↗

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 $\{(x,y)\in[0,1]^2:x\leq ye^{1-y}\}$.

math.CO↗

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.

math.CO↗

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.

math.CO↗

arXiv2020

Inverse problems for minimal complements and maximal supplements

Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to ask which subsets $C$ can arise as minimal complements for some $W$. We show that in a finite abelian group $G$, every non-empty subset $C$ of size $|C| \leq 2^{2/3}|G|^{1/3}/((3e \log |G|)^{2/3}$ is a minimal complement for some $W$. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for ``dual'' problems about maximal supplements.

math.CO↗