组合数学

For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real Schur roots of the associated root system. We study the two-coordinate projections $(c_v, c_w)$ of this configuration: when the difference $c_v - c_w$ is bounded the…

A Class-Uniformly Resolvable Design (CURD) is a resolvable design in which each parallel class has the same block structure. We study CURDS in which each parallel class contains one block of size $m$ and the remaining blocks have size $2$,…

For $k\geq 1$, we prove that \[ [q^n z^s]J_k(z,q)\geq 0, \qquad (n\geq 0,\ s\in\mathbb Z) \] for the normalized Jacobi triple product tails \[ J_k(z,q) = \frac{ \sum_{j=k}^{\infty}(-1)^{j-k} q^{\binom{j+1}{2}}(z^{-j}+\cdots+z^j)}…

The term tropical pseudonorm refers to a family of (not necessarily symmetric) gauge functions that arise in tropical or idempotent geometry. An important characteristic of these gauges is their invariance under translation by a constant…

We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries. Our main result treats the case of prescribed…

In this paper, we study totally disjoint diametral paths in simple connected graphs. A diametral path in a graph is a shortest path that connects two vertices whose mutual distance is equal to the diameter of the graph. Totally disjoint…

For a graph $G$, a proper $k$-coloring of $G$ is \emph{equitable} if the sizes of any two color classes differ by at most one. The \textsc{Equitable $k$-Coloring} problem asks, for a given graph $G$ and integer $k$, whether $G$ admits an…

Harborth's conjecture states that every planar graph has a crossing-free straight-line drawing in which every edge has an integer length. Kleber's strengthening asks for the vertices themselves to have integer coordinates. In this series of…

We show that the category of orthomodular posets is a full coreflective subcategory of the category of strong orthoposets, those orthoposets in which any two orthogonal elements have a join. This coreflection is obtained by building from a…

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$-systems. The main result is a proof of the Aljohani--Bamberg--Cameron conjecture which claims…

We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this…

We consider Subtraction Nim, where two players have exactly the same options, but which is partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. We consider the following example: Let $S$…

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo $p$ and $q$ are also square-free for relatively prime integers $p$ and $q$. Among such pairs $(p, q)$ with $p, q…

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call \emph{stotting}, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning…

Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. The proof is based on a bespoke container…

Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family by combining exact 0--1 ILP computations for the smallest cases with a constructive search procedure…

We derive the local and central limit theorems for the Stirling numbers of the second kind by elementary means, obtaining as corollaries effective asymptotic estimates for the Bell numbers and for the moments of the distribution. We also…

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if…

Given a group $G$, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of $G$ where each element of $G$ is included in the generating set independently at random with probability $p$. In this article, we…

In 2024, Ceballos and Chenevi{\`e}re introduced alt $\nu$-Tamari lattices, parameterized by a lattice path $\nu$ and an increment vector $\delta$, as a common generalization of $\nu$-Tamari and $\nu$-Dyck lattices. We study rowmotion on two…