Papers

A square matrix is said to be Hessenberg whenever each entry below the subdiagonal is zero, and each entry on the subdiagonal is nonzero. A Hessenberg matrix is called circular whenever the top-right corner entry is nonzero, and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. In 2022, Jae-ho Lee conjectured that a circular Hessenberg pair satisfies two relations called the tridiagonal relations. In the present paper, we prove Lee's conjecture. Our proof is not elementary.

The goal of the minimal model program for the Hilbert scheme of points on a surface aims is to describe the (stable) base loci of all divisors, their associated birational models, and the maps between them. We answer all of these questions for the Hilbert scheme of points on the blowup of the affine plane at the origin. The birational models are Brill-Noether loci in a larger Hilbert scheme, and nested variants thereof, and the wall-crossing maps are described as explicit projections. We also establish several new facts about the homogeneous coordinate ring of this Hilbert scheme, including finding the minimal set of line bundles whose sections generate the ring.

We construct an acyclic augmented chain complex of abelian groups whose entry in degree $n > 0$ is free on the set of noncrossing partitions of degree $n-1$ equipped with a $\{0, 1\}$-labelling of their gaps. The definition of the differential in this complex is related, via a restricted Leibniz rule, to the gap-insertion operad of Ebrahimi-Fard, Foissy, Kock, and Patras. We conjecture that this augmented chain complex is the linearisation of a polygraph presenting a self-dual model of the coherent walking $\omega$-equivalence constructed by the author, Loubaton, Ozornova, and Rovelli, and provide evidence for this conjecture.

Let $A$ be an arbitrary $2\times 2$ matrix. In \cite{Cisneros:PhD,Cisneros:I2x2M} I gave a formula for the trace of the $k$-th symmetric power of $A$ in terms of the anti-diagonal entries of $A^{k+1}$ and $A$. This was based on formulae that I found for the entries of the $k$-th power $A^k$ of the matrix $A$ in terms of its entries but I only sketched the idea of how I obtained such formulae. In this article I give the full proof of those formulae by counting some walks of length $k$ over the complete digraph of order $2$. I compare them with formulae for $A^k$ given by Mc Laughlin in \cite{McLaughlin:CIDnP2x2M} and by Williams in \cite{Williams:nthP2x2M}. This leads to combinatorial identities, in particular expressions for Fibonacci and Lucas numbers.

Recent work of Lam, Bychkov-Gorbounov-Kazakov-Talalaev, and Chepuri-George-Speyer gave a stratification of the totally nonnegative Lagrangian Grassmannian into electroid cells parameterized by cactus networks, paralleling Postnikov's stratification of the totally nonnegative Grassmannian by positroid cells. Electroid varieties arise as an algebro-geometric extension of electroid cells. The combinatorics of these varieties was studied by Lam in 2018. We build on this work and study the geometric properties of electroid varieties. In analogy to results of Knutson, Lam, and Speyer on positroid varieties, we show that electroid varieties are reduced, irreducible, regular in codimension one, compatibly Frobenius split, and form a stratification. We also show a decomposition of certain electroid varieties as a product of two electroid varieties. As a consequence, the grove measurement map that embeds electroid cells can be extended algebraically to embed an algebraic torus.

The flip graph of an origami crease pattern has the locally flat-foldable mountain-valley assignments as vertices, and an edge joins two of them that differ by a single face flip. A basic invariant of this graph is the degree sequence, which counts the vertices of each degree. On the $m\times n$ Miura-ori, this sequence is known to be a bivariate polynomial only for small degrees, each count obtained by a separate argument. This paper gives one uniform construction that expresses, for every degree $d$, the number of degree-$d$ vertices as a single symmetric polynomial in $(m,n)$ for all sufficiently large $m,n$. Its degree in each variable is $d-2$ unconditionally. Subject to a single degree bound, its total degree is $d-2$ as well, with top-degree part an explicit multiple of $m^{d-2}+n^{d-2}$ for $d\ge5$. The bound is proved here when the count splits into independent row and column factors, and open otherwise. The region is $m,n\ge\max(d-1,2)$. The polynomials are given in closed form through $d=10$, unconditional through $d=8$, where the degree bound holds in every case, and conditional on it beyond. Below this region the count departs from the polynomial. One step below, this departure has leading coefficient $-4$ times a Baxter number through $d=11$. Each such polynomial thus counts the Miura-ori's locally flat-foldable assignments admitting exactly $d$ single face flips.

We study the interplay between graded families of ideals in $\mathbb{K}$-domains and their associated convex regions. These regions, called Newton-Okounkov regions, arise naturally from graded families of ideals associated to a valuation with one-dimensional leaves. Our main focus is to compute asymptotic resurgence number of a pair of graded families of ideals. By combining techniques from Attouch–Wets topology and convex-geometric properties of Newton-Okounkov regions, we characterize the asymptotic resurgence number through containment relations between the pair of corresponding Newton-Okounkov regions.

Let $R(G,p)$ be the normalized rigidity matrix of a framework $(G,p)$ in $\mathbb R^d$, and let \[ L(G,p)=R(G,p)R(G,p)^{T} \] be the associated stiffness matrix. We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin. Our main result shows that, whenever $d\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. We further construct an infinite family of examples, given by regular polygons embedded in a two-dimensional subspace, for which the eigenvalue $n/2$ has multiplicity $2n-4$. Consequently, the multiplicity predicted in the conjecture is not correct in general. Our results reveal a dichotomy: the value of the second largest eigenvalue is universal, while its multiplicity is sensitive to the geometry of the underlying point configuration.

We show that there exists a constant $\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}^2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, \[ \max_{\lambda>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=\lambda\} \gtrsim \frac{|A|^2}{n^{1-\delta}}.\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erdős unit distance problem and the Elekes-Rónyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erdős. The existence of a set satisfying property (1) confirms a conjecture of Erdős from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

Let $G$ be a graph and let $\PF_t(G)$ denote the simplicial complex whose faces are vertex subsets whose induced subgraphs contain no path on $t$ vertices. These complexes encode a forbidden-subgraph condition as a family of allowed vertex subsets. In this paper, we study $t$-path-free complexes of complete multipartite graphs. Let \[ G=K_{n_1,\dots,n_m}, \qquad n_1\le\cdots\le n_m. \] We first obtain an explicit structural decomposition of $\PF_t(G)$ as a union of join complexes, together with an additional lower-dimensional size-truncation term. Using this decomposition, we show that for $t\le 2n_{m-1}-2$ the complex $\PF_t(G)$ is not sequentially Cohen-Macaulay, while for $t\ge 2 n_{m-1}-1$ it is vertex decomposable. Consequently, we obtain a complete characterization for complete multipartite graphs: $\PF_t(G)$ is vertex decomposable if and only if $t\ge 2n_{m-1}-1$. Equivalently, this is also exactly the range in which $\PF_t(G)$ is shellable and sequentially Cohen-Macaulay. We further analyze the topology via a Mayer-Vietoris spectral sequence: for complete bipartite graphs, we determine the full homotopy type as an explicit wedge of spheres in all cases.

Let $x$ be sufficiently large and \[ N(x)=\big|\bigl\{n\le x:n\ \text{is odd and }n\ne p+2^a+2^b \textrm{ with } p \text{ a prime and } a,b\in \mathbb{N}\bigr\}\big|. \] Motivated by Crocker's result \[ N(x)\gg \log\log x, \] Erd\H os repeatedly asked whether there is an absolute constant $c_0$ such that $N(x)>c_0x$. Pan \cite{Pan} proved in 2011 that \[ N(x)\gg x\exp\!\left( -C_0\frac{\log\log\log\log x}{\log\log\log x}\log x \right), \] where $C_0>0$ is an absolute constant. We improve on Pan's result by showing that, given any $\eta>0$, for all sufficiently large $x$, \[ N(x)\gg_\eta x\exp\left(-(4+\eta)\frac{\log\log\log x}{\log\log x}\log x\right). \]

We introduce the microscopic weighting, a canonical signed measure of mass one that can be associated to almost any finite metric space. The microscopic weighting is obtained as the small-scale limit of the weightings used to define the magnitude function. We give general criteria for its existence, proving in particular that every finite space of strictly negative type admits a microscopic weighting; this includes every finite subset of Euclidean or hyperbolic space and every finite tree. Heuristically speaking, the microscopic weighting distributes its mass as widely as possible across a space, assigning greater weight to sparse or outlying regions and emphasizing points on the boundary. Indeed, we show that on a finite space of negative type the microscopic weighting can be characterized (when it exists) as an optimizing measure for an energy integral determined by the distance function. Alternatively, it can be characterized in terms of the geometry of the Schoenberg embedding. Each of these interpretations also clarifies the information carried by the derivative of the magnitude function at zero. Though our main focus in this paper is on finite metric spaces, we lay the groundwork to extend the theory to compact subsets of Euclidean space. In that setting, we observe that the microscopic weighting must be understood as a distribution rather than as a measure.

After $n$ radial cuts of a circular cake, it is divided into $n$ slices. Call an adjacent pair of slices a portion. We exhibit an infinite sequence of cuts such that the ratio between the maximum and minimum sizes of a portion never exceeds 1.755. This improves on the trivial upper bound of 2, disproving a conjecture of Korsky.

We present the first constant-amortized-time algorithms for generating all length-$n$ necklaces and Lyndon words over a $k$-letter alphabet in colexicographic order, for arbitrary $k\geq 2$. Our approach introduces a novel class of words called quasinecklaces, which serve as an easily generated superset of necklaces through which all necklaces can be efficiently identified. We derive a formula for the number $Q_k(n)$ of length-$n$ quasinecklaces and show that $Q_k(n)$ is proportional to the number of length-$n$ necklaces, which is the key property needed to achieve constant amortized time. We also apply our results to efficiently generate a well-known de Bruijn sequence and efficiently generate necklaces and Lyndon words subject to a weight constraint.

Currently, among the open (unsolved) problems in number theory is the following: it is unknown what is the natural density of the sequence of Mersenne numbers in the set of natural numbers. In the paper, using methods of nonstandard mathematical analysis, we obtain the following equation: the natural density of Mersenne numbers (some infinitesimal value $e$) multiplied by the sum of the reciprocals of odd numbers (the infinitely large value {\omega} = 1 + 1/3 + 1/5 + ...) is equal to 1, or the equality $e$ = 1/{\omega} is true. In nonstandard analysis, the resulting infinitesimal numbers $e$ and 1/{\omega} are considered equivalent. We obtained this result by working with a two-dimensional matrix of non-negative integers, where odd numbers are separated from even ones by the Pepis-Kalmar pairing function.

We prove that for every fixed integer $r\geq 2$ and every $\varepsilon>0$, every sufficiently large finite uncrowded $(r+1)$-uniform hypergraph of maximum degree $\Delta$ has list chromatic number at most \[ (1+\varepsilon)\left(\frac{r\Delta}{\log\Delta}\right)^{1/r}. \] The proof is a semi-random list-coloring nibble carried out directly on the original hypergraph. We encode the remaining coloring problem by active edge-color constraints and control all residual sizes through a binomial degree bound. After the nibble reaches a sparse terminal state, the coloring is completed by a Rosenfeld-style counting argument.

What is the maximum number of points one can place in an $n \times n$ grid such that every Euclidean line contains at most $k$ points? For $k = 2$, this is the notorious no-three-in-line problem of Dudeney. In this paper, we resolve this problem for all other $k$ (and sufficiently large $n$). Namely, for $k \geqslant 3$ and sufficiently large $n$, we show that this maximum is exactly $kn$. To prove this, our key observation is that in the regime $k \geqslant 3$, the problem is dominated in a certain statistical sense by the influence of a small number of "heavy" lines with many grid points. We apply a result of Ehard-Glock-Joos on pseudorandom hypergraph matchings to construct a set of size $kn - o(n)$ with at most $k$ points on each heavy line, and then a crude deletion argument yields a no-$(k+1)$-in-line set of nearly the same size. Finally, we use a randomised switching procedure to complete the construction (building upon ideas of Simkin and Luria). Using similar ideas, we also address the no-four-on-a-circle problem of Erdős and Purdy. Namely, we prove the existence of a set of $2n - o(n)$ points in the $n \times n$ grid such that no four of these points lie on a circle or a line, improving on the previous construction of size $n - o(n)$ due to Dong and Xu.

We prove that the number of lozenge tilings of a certain triangular region $\mathcal{T}_n$ is given by the formula \[T_n=\prod_{\substack{1\leq a<b\leq 3n+2\\(a,b)\not=(n+1,2n+2)}}\left|1+\zeta^a+\zeta^b\right|^{1/3},\] where $\zeta=e^{2\pi i/(3n+3)}$. This answers a question of Ciucu and Krattenthaler, both by finding the exact formula and by explaining why $T_n$ has many prime factors. The proof reduces the lozenge tiling enumeration problem to evaluating the determinant of the bipartite adjacency matrix $M_n$ of the dual graph of $\mathcal{T}_n$, and then evaluates this determinant by diagonalising $M_n$.

We generalize Gessel's Formula for the number of Latin rectangles to partial Latin rectangles and non-attacking rook placements in a tridimensional chessboard. We also derive explicit short formulas for the generating series of the numbers of non-attacking rook placements on a chessboard with $2$ or $3$ levels. These series also count partial Latin rectangles with $2$ or $3$ rows. The results are obtained following methods developed by MacMahon and Gessel for counting Latin squares and Latin rectangles, by means of scalar products of multisymmetric functions.

We investigate metric dimension and the localization game for several families of directed analogues of strongly regular graphs and their generalizations, adapting a probabilistic method of Babai (1980) for bounding the size of resolving sets in undirected strongly regular graphs. We derive upper bounds on the localization number and metric dimension depending on the order of the graph and the maximum number of common out-neighbours for a pair of vertices. We consider normally regular digraphs, so-called "ordinary graphs", classes of Deza digraphs, divisible design digraphs, nearly doubly regular tournaments, and certain doubly regular team tournaments. In particular, for asymmetric normally regular digraphs on $n$ vertices, we show that these invariants are bounded above by $O(\sqrt{n} \log n)$, and improve this to $O(\log n)$ for a class of doubly regular team tournaments.

A measurable set $A\subseteq \mathbb S^{d-1}$ is called zero-sum-free if there are no $\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in A$ with $\boldsymbol{x}+\boldsymbol{y}+\boldsymbol{z}=\boldsymbol{0}$. Bukh asked whether every zero-sum-free measurable subset of $\mathbb S^{d-1}$, for $d\ge3$, has normalized surface measure at most $\frac{1}{2}$. He also pointed out that even the asymptotic behavior as $d\to\infty$ was unknown. We answer Bukh's asymptotic question by proving that every such set has normalized surface measure at most $\frac{\lfloor (d+1)^2/2\rfloor}{d(d+1)}=\frac{1}{2}+O\left(\frac{1}{d}\right).$ Since the lower bound $\frac{1}{2}$ comes from open hemispheres, this determines the asymptotic extremal density. By monotonicity, upper bounds in low-dimensional cases are especially important. We use a stability argument to improve the bound from $\frac{3}{5}$ to $\frac{71}{120}$ in dimensions $4$ and $5$.

We prove that the maximal normalized surface measure of a spherical set in d dimensions avoiding solutions to x + y + z = 0 approaches 1/2 as d goes to infinity. This gives a partial answer to a question of Bukh, who conjectured 1/2 to be the optimal bound for all d >= 3.

$2$-designs admitting a flag-transitive automorphism group $G$ with socle $PSL(2,q)$, where $q=p^{f}\geq 4$, are investigated in both the point-primitive and point-imprimitive cases. In the latter case, a complete classification is achieved, and three known examples occur, namely: the complementary designs of $PG(3,2)$ and $PG(3,4)$, and the $2$-$(36,8,4)$ design constructed by Devillers and Praeger in [14]. In the point-primitive case, apart from the Witt-Bose-Shrikhande linear spaces of even order $q$, $48$ sporadic examples are classified. Surprisingly, one of these numerical examples is the linear space with $v=496$ and $k=4$ admitting $P\Gamma L(2,2^{5})$ as a flag-transitive automorphism group, which was missing in the 1990 classification by Buekenhout et al. [7,36,12].

In this paper, we present a new construction of divisible design graphs with new parameters, obtained by plugging a difference set of a quotient group into a known construction of antipodal distance-regular graphs of diameter 3. Also, we show that in characteristic 2 the new divisible design graphs are Cayley graphs over an elementary abelian 2-group.

In 2017, McDiarmid and Yolov introduced the bipartite-hole-number $\widetilde{\alpha}(G)$ and proved that $\delta(G)\ge \widetilde{\alpha}(G)$ forces a Hamilton cycle. They also gave a sufficient condition for packing edge-disjoint Hamilton cycles, and asked whether this condition is sharp or can be relaxed. For integers $a,k\ge 2$, let $f(a,k)$ be the least integer $d$ such that every graph $G$ on at least three vertices with $\widetilde{\alpha}(G)\le a$ and $\delta(G)\ge d$ contains $k$ pairwise edge-disjoint Hamilton cycles. We prove that $f(a,k)=\Theta\left(a+k+\frac{ak}{\log(k+2)}\right).$ The upper bound uses a deletion lemma for the bipartite-hole-number together with the McDiarmid–Yolov Hamiltonicity theorem and a greedy packing argument. The lower bound is obtained from three extremal constructions, the logarithmic one using a sparse random auxiliary graph with no prescribed bipartite hole.

Generically the rigidity of bar-joint structures admits combinatorial characterisations in the Euclidean plane and, more generally, for frameworks on the sphere and the torus. The remaining case of compact surfaces of genus at least two has remained open. Using the hyperbolic geometry of their universal covers, we develop a theory of infinitesimal rigidity for frameworks on compact surfaces of genus at least two. By the uniformisation theorem, every such surface is a quotient of the hyperbolic plane by a surface group, allowing frameworks on the surface to be represented as infinite symmetric frameworks in the hyperbolic plane. Encoding the symmetry through gain graphs, we prove that infinitesimal rigidity is determined entirely by finite combinatorial data. Specifically, a framework is generically rigid if and only if its associated gain graph contains a spanning (2,3,1,0)-gain tight subgraph. This yields the first combinatorial characterisation of generic rigidity for frameworks on compact surfaces of genus at least two.

In a previous work, we defined the complex of cluster parking functions. On one side, they encode the type-refined enumeration of faces of the cluster complex, and on the other side, they have a reduced homology which is isomorphic to (ungraded) diagonal coinvariants. The goal of this work is to take into account the underlying dihedral symmetry. We thus have a product of a dihedral group and a symmetric group (there is a precise conjecture in the case of other finite Coxeter groups, but we focus on symmetric groups because of technicalities about diagonal coinvariants beyond this case). Under the action of the product group, the reduced homology of cluster parking functions is conjecturally isomorphic to diagonal coinvariants up to tensoring by a sign character of the dihedral group. This isomorphism can be reformulated as a dihedral sieving phenomenon. The main technical contribution is the definition of the dihedral automorphism group of cluster parking functions, and we discuss various features of the reduced homology character and its conjectural connection with diagonal coinvariants.

For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ of the edge ideal of a graph $G$ is generated by the squarefree monomials supported on $q$-matchings of $G$; it is the Stanley–Reisner ideal of the complex $\Delta_q(G)=\{F\subseteq V(G):\nu(G[F])<q\}$, where $\nu$ denotes matching number. We prove a general formula for the matching number of an arbitrary graph join, \[ \nu(G\ast H) = \min\Big(\nu(G)+|V(H)|,\ \ \nu(H)+|V(G)|,\ \ \Big\lfloor\tfrac{|V(G)|+|V(H)|}{2}\Big\rfloor\Big), \] via the Tutte–Berge formula, and use it to decompose $\Delta_q(G\ast H)$ for arbitrary graphs $G,H$. Specializing to the wheel graph $\mathcal{W}_n = \mathcal{C}_n\ast\mathcal K_1$, we determine the Krull dimension and height of $R/I(\mathcal{W}_n)^{[q]}$ exactly for all $n\ge 3$, $1\le q\le\lfloor n/2\rfloor$, and – combining our matching-number computations with a recent Tutte-type Cohen-Macaulayness criterion of Ficarra and Moradi – prove that at the top squarefree power $q=\nu(\mathcal{W}_n)=\lceil n/2\rceil$, the ideal $I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}$ is literally the squarefree Veronese ideal, so that $R/I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}$ is Cohen-Macaulay with \[ {\rm dim} = {\rm depth} = {\rm reg}\big(R/I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}\big) = 2\Big\lceil\frac n2\Big\rceil-1. \] This resolves all four classical invariants at the top power, and confirms there the pattern depth$(R/I(\mathcal{W}_n)^{[q]}) = 2q-1$ that our computational data (now extended to $n\le13$, every valid $q$) suggests holds throughout. We prove a general depth formula for squarefree powers of cone graphs, via a Betti-splitting exact sequence, that reduces this pattern to two more tractable statements about the underlying cycle alone; both are verified computationally in every case checked but left open in general.

We show that given completely multiplicative functions $f_1,\dots,f_d$ taking values in the unit circle, there exist Pythagorean triples (i.e., integer solutions to $x^2+y^2=z^2$) with $f_i(x),f_i(y),f_i(z)$ all arbitrarily close to $1$ for all $i$. This is a new special case of the conjecture that any finite colouring of $\mathbb{N}$ has a monochromatic Pythagorean triple. Our proof combines vanishing averages for aperiodic functions with concentration estimates for pretentious functions. A similar proof is applied to obtain the analogous statement for more general equations of the form $ax^2+by^2=cz^2$ whenever $a,b,c$ are perfect squares satisfying the Rado's condition.

Steinerberger curvature encodes the global distance geometry of a graph through an equilibrium measure. In this paper, we derive explicit curvature formulas for undirected Cayley graphs of dihedral groups $D_n$ and generalized quaternion groups $Q_{4m}$. We then extend Steinerberger curvature to strongly connected simple digraphs by introducing in-curvature and out-curvature, reflecting the asymmetry of directed distances. For these directed curvatures, we establish structural properties, including negativity criteria and a permutation relation between in- and out-curvature. Our main results are directed analogues of the Bonnet–Myers, Cheng and Lichnerowicz theorems, together with reverse Bonnet–Myers inequalities for directed diameter and out-radius, and an upper bound for in-radius in terms of total curvature.