Papers

Given an oriented graph $\vec{G}$ and a subset of vertices $X \subseteq V(\vec{G})$, the inversion of $X$ is the operation that reverses the orientation of every arc with both endpoints in $X$. For a simple graph $G$, the inversion diameter $\operatorname{diam}(I(G))$ is the maximum distance between two orientations of $G$ under inversions of vertex sets. We prove the sharp bound \[ \operatorname{diam}(I(G))\le 2\chi_a(G)-2, \] where $\chi_a(G)$ is the acyclic chromatic number. Consequently, every planar graph has inversion diameter at most $8$, improving the previously known bound $12$. Using strong-degeneracy arguments, we also obtain upper bounds $7$, $5$, and $4$ for planar graphs of girth at least $4$, $5$, and $6$, respectively.

Let $\pi(H)$ be the Turán density of an r-uniform hypergraph $H$ and let $H_k^r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\le k\le r+1$. Sidorenko~(JCT-B, 2024) proved that $\pi(H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to\infty$ and $\pi(H_k^r)\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $\pi(H_3^r)\ge cr^{-2}\sqrt{\log r}$ for some constant $c>0$. In this article, we prove the following results. \begin{itemize} \item For any fixed $\varepsilon>0$, there is a constant $c_\varepsilon>0$ such that $$\pi(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)^{2+\varepsilon}}.$$ %$\pi(H_3^r)\ge 1/(r(\log r)^{2+o(1)})$. Together with the known upper bound $\pi(H_3^r)\le1/r$, this implies $\pi(H_3^r)=r^{-1+o(1)}$. \item For every $3\le k\le r+1$, let $s=\min\{k-2,r-k+2\}$. Then \begin{equation*} 0\le \frac{k-2}{r}-\pi(H_k^r) \le \frac{128}{r}\left(\sqrt{s\log\frac{er}{s}}+\log\frac{er}{s}\right). \end{equation*} This estimate yields several asymptotically sharp results for $\pi(H_k^r)$. For example, $\pi(H_k^r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}

We consider the following question: if two lattice paths in the same set $\mathcal{D}(a,b)$ have the same Lagrange number, must their band graphs be isomorphic? We exhibit two explicit lattice paths in $\mathcal{D}(17,9)$ with the same Lagrange number but non-isomorphic associated band graphs, thereby giving a counterexample to this question.

We count Schreier sets that satisfy a neighborhood condition, including $k$-clustered, $k$-consecutive-free, $k$-neighbored, $k$-isolated, and closed under integral $2$-averages. For the first four conditions, we determine the initial counts and prove linear recurrence relations. For the last condition, we prove a recurrence that involves the divisor counting function.

For admissible integers $n,\alpha,k$, let $f(n,\alpha,k)$ be the minimum number of edges in a $k$-connected graph of order $n$ and independence number $\alpha$. A conjecture of Bougard and Joret predicts that $f(n,\alpha,k)=\lceil nk/2\rceil$ when $n\leq k\alpha$, under the assumptions $n\geq2\alpha$, $n\geq\alpha+k$, $\alpha\geq2$, and $k\geq3$. We disprove this prediction, determine $f(n,\alpha,k)$ throughout the boundary $n=\alpha+k$, and characterize every extremal graph on that boundary. In particular, for every $k\geq4$, \[ f(2k-1,k-1,k)=k^2-1, \] whereas the conjectured value is $k^2-\lfloor k/2\rfloor$. The extremal graphs in this family are precisely $\overline K_{k-1}\join T$, where $T$ is an arbitrary tree of order $k$. The smallest-order failure has parameters $(n,\alpha,k)=(7,3,4)$, and no admissible counterexample has smaller order.

Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb F_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots+q^k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over $\mathbb F_2$ whose $k=3$ kernels cover $\mathbb F_2^7$; in particular, $C_3(2)\le 13<15$. For a general covering family of size $N=q^k+S$ and $\mathbb F_q$-linear rank $d$, we prove $S\gg d^{2/3}\left(\frac{\log(2q)}{\log(eNq^k/S)}\right)^{2/3}$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q^k+c_kq^{2/3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields $\liminf_{q\to\infty}(C_2(q)-q^2)/q\ge \widetilde c_2$, where $\widetilde c_2$ is an explicit one-variable variational constant with numerical value $\widetilde c_2=0.5829944375\ldots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q^k+(1/2-o(1))q^{k-1}$.

By a short adaptation of an argument of Singerman, we show that for every prime power $q$, every orientably regular map (or hypermap) whose orientation-preserving automorphism group is isomorphic to PGL(2,q) is reflexible.

For every $d\geq3$ we construct a connected graph of order $2\binom{3d+1}{d}$ that is locally $K(2d+1,d)$. Fix a block $A$ and put $a'=\min(|A|,3d+1-|A|)$. Among the loopless binary voltages on the fixed labeled base $K(3d+1,d)$ that depend only on the intersection types with $A$, the local-neighborhood-preserving assignments form, modulo gauge, an $\mathbb F_2$-space of dimension $[q^{a'-5}]\binom{d}{3}_q$, with explicit canonical coordinates. Adjacent-line rigidity holds on $2d+2\leq n\leq3d$: every loopless fixed-block type-invariant local-neighborhood-preserving binary voltage is gauge trivial. Dropping type invariance over the fixed labeled $K(10,3)$, we classify all local-neighborhood-preserving binary voltage assignments and find that $H^1(M_3(10);\mathbb F_2)$ has dimension 42. Under base relabeling by $S_{10}$ these classes form 1,245,395 orbits, of which 1,245,394 consist of connected covers; an explicit class has orbit size 126 and stabilizer $S_5\wr C_2$. The proofs combine a triangle-voltage criterion and cohomology with a simplicial collapse and Burnside enumeration.

We define a new matrix associated with a hypergraph. A study of this matrix is carried out, in particular the study of its spectrum, which shows the relevance of the construction. We demon- strate that this adjacency matrix is characterisitic of the hypergraph, that is to say, two isomorphic hypergraphs have similar matrices. Furthermore, starting from this matrix, we can reconstruct the hypergraph. Starting from this matrix, we introduce new graphs associated with the hypergraph.

Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an $S^1$-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an $S^1$-flow. We then develop some reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit $S^1$-flows, respectively.

In this paper, we present a general method of deducing the determinantal expressions for a polynomial and its derivative. As illustrations, we provide three determinantal expressions for the derivative of the Eulerian polynomial. Using a functional equation discovered by Gessel,we also establish the determinantal expressions for the second-order Eulerian polynomial and its derivative.

In 2016, Reiher's clique density theorem determined the minimum number of copies of $K_t$ in a graph with a prescribed edge density. In this paper, we investigate its local version and prove a local clique density theorem in $H$-free graphs as follows. For integers $r$ and $t$ with $2\leq t\leq r-1$, any $r$-chromatic graph $H$, any real numbers $\gamma$ and $\alpha$ with $\frac{t-2}{2(t-1)}\leq\gamma\leq \frac{r-2}{2(r-1)}$ and $0\leq\alpha\leq 1$, we determine the maximum value $\beta:=\beta(r,t,\alpha,\gamma)$ such that for every $n$-vertex $H$-free graph $G$ with at least $\gamma n^2$ edges, every $\lceil\alpha n\rceil$-vertex subset in $G$ contains at least $(\beta-o(1))n^{t}$ copies of $K_t$. In particular, when $H=K_r$, every $\lceil\alpha n\rceil$-vertex subset contains at least $\lfloor\beta n^t\rfloor$ copies of $K_t$, which is an exact bound. For suitable choices of $\alpha$ and $\gamma$, namely, those for which all part ratios in the corresponding extremal construction are rational, this bound is attained for infinitely many values of $n$.

Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^2$, then the alternating areas obtained by ordering $B$ assume at least $\min(p,n-1)$ distinct values. Its proof is a short contraction induction: contract a suitable independent pair, replace the contracted vector in both orders, and apply Cauchy–Davenport. A polynomial relative-subsum theorem and a sharp representation-rigidity lemma then turn this local growth into a uniform spread bound. Combined with the standard product-one criterion for $H_{p^3}$, the spread bound yields the upper bound; the usual sequence $x^{p-1}y^{p-1}v^{p-1}$ gives the lower bound.

We study the random induced-$P_4$-free graph process. Let $e_1,\ldots,e_N$, where $N=\binom{n}{2}$, be a uniformly random ordering of the edges of $K_n$. Starting from the empty graph $G_0$, we add $e_{m+1}$ whenever $G_m+e_{m+1}$ contains no induced $P_4$, and otherwise leave the graph unchanged. We show that the terminal graph is a trivially perfect graph and we describe the structure and distribution of the connected components of the terminal graph $G_N$. Consequently, we derive the limiting values of several natural graph parameters. In particular, the terminal graph $G_N$ has $\Theta(n)$ edges.

A vertex-coloring of a graph is centered if every connected subgraph has a vertex with a unique color. A vertex-coloring of a graph is linear if every path in the graph has a vertex with a unique color. Let $\chi_{\mathrm{cen}}(G)$ and $\chi_{\mathrm{lin}}(G)$ be the minimum number of colors in a centered (resp. linear) coloring of $G$. We present a family of graphs witnessing that if $f$ is a nondecreasing function such that $\chi_{\mathrm{cen}}(G) \leq f(\chi_{\mathrm{lin}}(G))$ for every graph $G$, then $f(k) = \Omega(k^2 / \log k)$. The construction was found by OpenAI's GPT-5.6 Sol Pro.

In a graph $G$, a set of edges $F$ is called a forcing set if there exists a unique perfect matching $M$ such that $F \subseteq M$. Similarly, a set of edges $A$ is called an anti-forcing set if the graph with edge set $ E(G)\setminus A$ has a unique perfect matching. It is known that, given a bipartite graph $G$ of maximum degree~$3$ and a perfect matching $M$, the problem of deciding whether there exists a forcing set of size at most $k$ for $M$ is NP-complete. Moreover, given a bipartite graph $G$ of maximum degree~$4$ and a perfect matching $M$, the problem of deciding whether there exists an anti-forcing set of size at most $k$ for $M$ is NP-complete. Furthermore, given a bipartite graph of maximum degree~$5$, the problem of deciding whether there exists a perfect matching $M$ that can be made unique by a forcing set of size at most $k$ is also NP-complete. In contrast, the computational complexity of deciding whether there exists a perfect matching $M$ that can be made unique by an anti-forcing set of size at most $k$ is not known, even for general graphs. In this paper, we show that all of these problems remain NP-complete even when restricted to bipartite graphs of maximum degree~$3$.

A finite group has the weak Erdos-Ko-Rado property if all of its transitive permutation actions have the EKR property. We characterize this property in terms of normal subgroups and chief factors. More precisely, we introduce a local intersection density and establish a normal-extension criterion which reduces the weak EKR property to difference-set conditions on the elementary abelian chief factors and the linear groups induced on them. For chief factors of rank one the condition is automatically satisfied, and for chief factors of rank two, this condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces. In the second part of the paper, we solve an open problem by determining the possible intersection densities of transitive permutation groups with a prescribed point stabilizer. We prove that, for every finite group $H$ of order $m\geq 4$ and every integer $n\geq m$, there exists a faithful transitive permutation group with point stabilizer isomorphic to $H$ and intersection density $n/m$.

We study Carlet's cyclic-additive conjecture for the Kasami almost perfect nonlinear (APN) function $F(x)=x^{4^k-2^k+1}$ on $GF(2^n)$, $\gcd(k,n)=1$: for the $2^{n-1}$-element set $\Delta=\{F(b)+F(b+1)+1: b\in GF(2^n)\}$ and all distinct nonzero $v_1,v_2\in GF(2^n)$, \[ \bigl|\{(x,y,z)\in\Delta^3 : v_1x+v_2y+(v_1+v_2)z=0\}\bigr| \;=\; 2^{2n-3}. \] This exact triple-count condition was first formulated by Carlet in his 2018 cyclic-additive difference-set framework; the Kasami instance was subsequently posed as an open problem at the NSUCRYPTO 2019 cryptographic olympiad, whose individual proposer was not publicly disclosed. We prove the conjecture for $k\bmod n\in\{1,2,n-2,n-1\}$, in particular a complete proof for $k=2$ ($d=13$) via a quadratic-form theory and an exact root-count reduction, and we verify it exhaustively by computer for every admissible $(n,k)$ with $n\le13$.

Teng and Tian proved the following result. Let $k\ge 2$ and $t\ge 3$, and let $G$ be a $k$-connected graph of order $n$. If $n\ge 6k+1$ and $\delta(G)\ge \lceil(n+6)/2\rceil$ when $t=3$, while $n\ge 6k+7t-17$ and $\delta(G)\ge \lceil(n+t+2)/2\rceil$ when $t\ge 4$, then, for any two distinct vertices $u,v$ and every integer $s$ with $1\le s\le t$, there exist $s$ internally vertex-disjoint $(u,v)$-paths $P_1,\dots,P_s$ whose union spans $G$ and such that $G-E(P_1\cup\cdots\cup P_s)$ is $k$-connected. They asked whether the minimum-degree condition could be lowered to $\delta(G)\ge \lceil(n+t)/2\rceil$ for every $t\ge 3$. We answer this question affirmatively and further reduce the required order to $n\ge \max\{6k+9-3t,\,2k+t+3\}$.

Let $\mathcal{C}_{[4,r]}$ be the family of cycles $\{C_4, \dots, C_r\}$. A graph $G$ is said to be $\mathcal{C}_{[4,r]}$-saturated if $G$ does not contain a copy of cycle $C_i$ for $4\le i\le r$, but the addition of any edge $e\notin E(G)$ creates at least one copy of $C_i$ for $4\le i\le r.$ The saturation number $sat(n, \mathcal{C}_{[4,r]})$ is the minimum number of edges in an $n$-vertex $\mathcal{C}_{[4,r]}$-saturated graph. In 2025, Ma determined that $sat(n, \mathcal{C}_{[4,5]})=\lceil \frac{5n}{4} - \frac{3}{2} \rceil$, and conjectured that for any $r \ge 5$, $sat(n, \mathcal{C}_{[4,r]}) = \lceil\frac{5n}{4} - \frac{3}{2} \rceil$ holds for large $n$. In this paper we prove that $sat(n, \mathcal{C}_{[4,r]}) \le \lceil\frac{5n}{4} - \frac{r+1}{4}\rceil$ for $n \ge r+1$, which disproves Ma's conjecture for $r\ge 6.$ For $r=6,$ we determine that $sat(n, \mathcal{C}_{[4,6]})=\lceil\frac{5n}{4}-\frac{7}{4}\rceil.$ Keywords: Saturation graphs; Saturation number; Cycles; Edge minimization

The codegree Turán density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $\pi_{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Turán density if and only if it is layered and has zero uniform Turán density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies \[ \pi_{\mathrm{co}}(F)\ge q_{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, \] which implies any $k$-graph with zero $\ell$-degree Turán density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.

For a divisor on $\overline{M}_{0,n}$, we introduce the polytope of its effective boundary expressions. We establish structural properties of these polytopes under the forgetful maps of $\overline{M}_{0,n}$ forgetting marked points, and give equivalent graph-theoretic descriptions. We compute these polytopes for several families of divisors. For psi-classes and their pullbacks by forgetful maps, we show that the polytopes are unimodular simplices. For the log-canonical class and its modifications by psi-classes, we prove that the nonnegative parts of the corresponding polytopes recover spanning forest polytopes and the subtour elimination (Held–Karp relaxation) polytope of the symmetric traveling salesman problem. As an application, we obtain a Minkowski-like decomposition of the subtour elimination polytope into simplices. Finally, for symmetric level-one $\mathfrak{sl}_p$ conformal block divisors, we show that the defining inequalities are local Turán bounds and the $0/1$-points are balanced Turán graphs. Moreover, for $p=2$ and $p=n/2$, these polytopes recover the perfect matching and fractional perfect matching polytopes.

Let $M$ be a simple matroid and let $B(M)$ be the graded Möbius algebra of its lattice of flats. The ordered-basis weights of flats define an inner product for which the adjoints of atom multiplication become ordinary coordinate derivatives under the basis-polynomial realization. From this, we construct a canonical global lowering operator $D_\beta$ which acts as ordinary differentiation on a canonical “radial” copy of a truncated polynomial algebra. Allowing both $D_\beta$ and the coordinate derivatives to act produces a graded cyclic module with Hilbert series \[ H_{\beta,M}(q)=\sum_{k=0}^r h_k^\beta(M)q^k. \] We give examples of matroids with the same Derksen $\mathcal G$-invariant and the same classical apolar Hilbert series but different $H_\beta$. Hence $H_\beta$ cannot be the restriction to simple matroids of a valuative matroid invariant. We conjecture that $H_\beta$ is log-concave and top-heavy in differential degree. For the generalized theta family containing Larson's counterexample to Whitney log-concavity, we compute the first four coefficients and prove the critical log-concavity inequality. Exact computation verifies both conjectures for all $950$ simple matroids on eight elements.

The two-dimensional bandwidth minimization problem (2DBMP) seeks an injective embedding of a guest graph into a square grid that minimizes the maximum Manhattan distance over its edges. Heuristic methods can provide strong upper bounds, but these bounds do not by themselves certify optimality. We present an efficient exact SAT-based approach for 2DBMP that incrementally searches for the minimum feasible bandwidth and certifies optimality through satisfiability and unsatisfiability results. On the standard $\lceil\sqrt n\rceil \times \lceil\sqrt n\rceil$ host grid, under a 3600 s time limit, the proposed SAT approach certifies optimal bandwidths for 41 of 43 Regular instances and 42 of 93 Harwell–Boeing instances, achieving substantially broader optimality certification within the 3600 s time limit than a previous exact approach evaluated with a 72-hour time limit. In addition, it certifies three bandwidth values that improve all previously published comparison values considered in this study and establishes all three as optimal. We further evaluate the approach on alternative host geometries, namely $2\times\lceil n/2\rceil$ and $n\times n$ grids, to assess its effectiveness beyond the standard host. Overall, the results demonstrate that the proposed SAT approach provides an effective exact method for the small- and medium-sized benchmark instances considered in this study, with fewer than 400 vertices, while heuristic methods remain important for larger and more challenging instances.

Let $\Sigma=(G,\sigma)$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(\Sigma)$ and $s^{-}(\Sigma)$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected graph $G$ of order $n$ satisfies \[ \min\{s^{+}(G),s^{-}(G)\}\ge n-1. \] Liu and Ning~\cite{LiuNing2023} published a wide-ranging paper entitled “Unsolved Problems in spectral graph theory", and this conjectures were placed first in their list of such problems. We prove that every signature $\sigma$ of a connected graph $G$ satisfies the sharp bound \[ s^{+}(\Sigma)\le 2m-n+1. \] For the all-positive signing this gives $s^{+}(G)\le 2m-n+1$, whereas for the all-negative signing it gives $s^{-}(G)\le 2m-n+1$. Since $s^{+}(G)+s^{-}(G)=2m$, these two special cases imply the square-energy conjecture; the present theorem is stronger in scope because the same bound holds for every signing of $G$. Applying the theorem to the negation $-\Sigma$ also yields \[ s^{+}(\Sigma)\ge n-1. \] Both bounds are sharp. The proof is based on a doubly nonnegative matrix inequality. We also shorten the proof of that inequality by replacing its final case distinction with a fixed convex combination.

We prove the nonexistence of eight circulant weighing matrices from the remaining table of orders at most $200$ and weights at most $100$. The proofs combine contraction, character evaluation on the kernel of a contraction, multiplier methods, and exact finite computations. For $CW(105,36)$, the contracted matrix is unique up to equivalence. Applying a nonprincipal character of the $C_3$ kernel gives an element over the Eisenstein integers; reduction modulo $1-\omega$ gives a word in a ternary cyclic code of length $35$, and exact enumeration rules out every required Eisenstein-unit lift. For $CW(140,36)$, the real-valued character $Y\mapsto-1$ of the $C_4$ kernel is incompatible with the same contracted class. For weight $64$, the faithful character $Y\mapsto i$ of a $C_4$ kernel first gives an element of $\mathbb{Z}[i][C_m]$; a generalized multiplier then forces constancy on multiplication-by-$2$ orbits, and exact correlation calculations eliminate orders $140$, $180$, and $196$. The three weight-$49$ cases are settled by the ordinary prime-power multiplier, with contraction where needed. Consequently none of $CW(105,36)$, $CW(140,36)$, $CW(116,49)$, $CW(120,49)$, $CW(192,49)$, $CW(140,64)$, $CW(180,64)$, and $CW(196,64)$ exists.

It is well known that, for each classical pattern $\tau$ of length 3, the number of $\tau$-avoiding permutations of length $n$ is the $n$th Catalan number, and numerous bijections between different length-three avoidance classes have been constructed and studied. In this paper, we refine this classical problem by studying Wilf equivalence among permutations that simultaneously avoid a classical pattern of length three and a flat partially ordered pattern. Partially ordered patterns (POPs) provide a flexible framework for encoding families of classical permutation patterns. For $\ell\geq 3$ and $1\leq x\leq\ell$, let $P_{\ell,x}$ be the length-$\ell$ POP in which the entry at position $x$ is required to be smaller than all the other entries, while no relations are imposed among the remaining entries. Such POPs are called flat POPs. We classify the Wilf equivalences among all pairs $(\tau,P_{\ell,x})$, where $\tau$ is a classical pattern of length three. For every $\ell\geq4$, the resulting $6\ell$ pairs form exactly $2\ell-1$ Wilf equivalence classes, while the exceptional case $\ell=3$ gives four classes. Our proofs combine the derivation of explicit formulas and recurrence relations with the construction of bijections. Moreover, we introduce novel prime-divisor arguments to distinguish the remaining candidate classes, reducing the problem to showing that a certain Diophantine equation has no solutions for $\ell\ge 3{,}274$, where the bound $3{,}274$ is not claimed to be sharp. Finally, by extending our work on POPs, we resolve a conjecture of Qiu and Remmel concerning the distribution of quadrant marked mesh patterns on 132-avoiding permutations and correct an error in their paper that is crucial to the proof.

A theorem of Dougherty–Jackson–Kechris states that any equivalence relation generated by a single Borel function is hypersmooth. A well-known open problem is whether this can be generalized to equivalence relations generated by countable families of pairwise commuting Borel functions. We give an affirmative answer in the case where the functions are bounded-to-one. This generalizes the theorem of Gao–Jackson on Borel actions of countable abelian groups.

We study the Ehrhart theory of half-open $d$-dimensional integer parallelepipeds $\Pi$. Although the lattice-point count $t\Pi\cap \Z^d$ is known to be simply $\vol \Pi t^d$ for positive integer $t$, the corresponding counting function for arbitrary real dilations $t$ has subtle, nontrivial periodic structure. We give explicit formulas for this real Ehrhart quasi-polynomial, and more generally for all the discrete moments of the real dilates of $\Pi$: $\sum_{p\in t\Pi\cap\mathbb Z^d}\langle p,z\rangle^m$. The formulas are expressed in terms of Barnes polynomials and polytope Dedekind sums, which encode the periodic lattice flow of translated integer lattices on the flat torus determined by $\Pi$. Our approach develops further the study of polytope Dedekind sums, introduced recently in \cite{Robins2026}. In particular, we obtain novel identities for polytope Dedekind sums by using iterated discrete derivatives. Moreover, we show that the Ehrhart quasi-coefficients of $L_\Pi(t)$ are precisely alternating sums of polytope Dedekind sums. Finally, we give an Ehrhart-type reciprocity law relating $L_{\Pi}(t)$ at negative arguments to the lattice-point count of the `opposite' half-open parallelepiped.

Let $H$ be a unimodular hypergraph with cover ideal $J(H)$. We prove that the local $v$-numbers of $J(H)^t$ are linear in $t$ for all $t\ge1$. We further show that the global $v$-number of $J(H)^t$ is linear in $t$ for all $t\ge n-1$. Finally, we prove that the global $v$-number of the powers of the cover ideal of any tree is linear in $t$ for all $t\ge1$.