The Ehrhart series of magic squares of order seven

Let $\mathrm{IMS}_n(m)$ denote the number of $n\times n$ nonnegative integer matrices whose row sums, column sums, and two main diagonal sums are all equal to $m$. We determine the Ehrhart series $F_7(q)=\sum_{m\ge 0}\mathrm{IMS}_7(m)q^m$ as a reduced rational function. The denominator has degree $373$ and cyclotomic factors of order at most $15$; the numerator is a palindromic polynomial of degree $366$ with nonnegative integer coefficients.   Using the SimpCone decomposition, the associated polytope is represented as a sum of $166$ million signed simplicial cones. The LRQC evaluator computes their generating functions over finite fields; a typical cone requires only one or two quotient characters, and the cost per character is nearly linear in the truncation degree $T$. An explicit common denominator together with Ehrhart reciprocity reduces the rational reconstruction to the prefix up to $T=1256$, while an explicit counting bound supplies the coefficient bounds needed for deterministic lifting from the prime fields to $\mathbb Z$. This prefix is independently computed for the whole cone family in eight prime fields. Exact Chinese remaindering lifts the verified residues to equality over $\mathbb Z$, the finite-prefix criterion proves the rational identity, and exact polynomial gcds prove that the displayed denominator is reduced.


Critical-exponent stratification and inverse realization on biregular trees

For free type-preserving discrete actions on the biregular tree $\mathcal T_{r+1,s+1}$, we stratify the critical-exponent spectrum by quotient complexity. The unrestricted spectrum is the full interval $[0,\frac12\log(rs)]$, whereas the finitely generated spectrum is countable and dense and is encoded by the Hashimoto radii of finite typed cores. At fixed rank, finitely many typed kernels parametrize all values, and every nonzero accumulation belongs to a lower-rank stratum. Rank two admits a complete effective inverse classification through the figure-eight, theta, and dumbbell polynomial families.


An $O(t\log^2 t)$ Bound for $k$-Connected Subgraphs in Dense $K_t$-Minor-Free Graphs

Delcourt and Postle reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices. An important theorem in their proof process asserts that every sufficiently dense $K_t$-minor-free graph contains a small, highly connected subgraph. In this paper, we show that such a subgraph can be chosen to be smaller. More precisely, there exists an integer constant $C \ge 1$ such that, for all integers $t \ge 3$ and $k \ge t$, every $K_t$-minor-free graph $G$ with $d(G) \ge Ck$ contains a nonempty $k$-connected subgraph $H$ satisfying $v(H) \le C^2 t\log^2 t$. Thus the structural bound improves from $O(t\log^3 t)$ to $O(t\log^2 t)$, and the graphs occurring in the reduction have order $O(t\log^3 t)$ rather than $O(t\log^4 t)$.


We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are:   (1) $Δ$ skeleton-E-chordal $\Rightarrow$ $Δ^\vee$ vertex-decomposable $\Rightarrow$ $Δ$ skeleton-clique-chordal. Moreover, for subflag complexes, $Δ$ skeleton-E-chordal $\Longleftrightarrow$ $Δ^\vee$ vertex-decomposable. (For $d=1$ this boils down to ``$G$ chordal $\Longleftrightarrow$ $G^\vee$ vertex-decomposable'', a result closely related to Fröberg's theorem.)   (2) For subflag complexes, $Δ$ is skeleton-E-chordal $\Longleftrightarrow$ it splits as $Δ= Δ_1 \cup Δ_2$, with each $Δ_i$ a skeleton-E-chordal induced subcomplex of $Δ$, and with $Δ_1 \cap Δ_2$ a complex whose $1$-skeleton is a clique. (This generalizes ``$G$ chordal $\Longleftrightarrow$ $G$ splits as a union of chordal graphs that intersect in a common clique'').   (3) $Δ$ skeleton-E-chordal $\Longleftrightarrow$ every nonempty induced subcomplex of $Δ$ has a skeleton-E-simplicial vertex. (Generalizes ``$G$ chordal $\Leftrightarrow$ every nonempty induced subgraph has a simplicial vertex''.)   (4) $Δ$ underclosed $\Rightarrow$ $Δ$ skeleton-weakly-chordal and weakly-closed. (Generalizes ``$G$ interval $\Rightarrow$ $G$ chordal and co-comparability''.)   (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)


Computations of $\tilde{A}_2$ Bruhat intervals via shadows

We develop an explicit geometric and algebraic description of shadows in the affine Coxeter complex of type $\tilde{A}_2$, by introducing a coordinate system based on a decomposition of the complex into tunnels and channels. We also provide an algorithm for converting arbitrary reduced words into coordinates. Using this framework, we identify geometric symmetries of shadows and show that they are governed by the underlying channel structure. This allows us to derive explicit, piecewise formulas for the cardinality of shadows in $\tilde{A}_2$, depending on the parity of the coordinates. Furthermore, we establish a simple criterion for shadow membership via a counting function that detects admissible positions within channels. These results provide a concrete and computationally effective description of shadows in affine type $\tilde{A}_2$, bridging the gap between combinatorial definitions and geometric realisations.


An Isodiametric Theorem and Lattice Diameter-Perfect Codes in $A_3$

The root lattice $A_n$, equipped with its graph distance (equivalently, one half of the ambient $\ell_1$ metric), is isometric to $\mathbb{Z}^n$ with the asymmetric Manhattan metric. We study two extremal problems in this space -- the isodiametric problem, i.e., determining the maximum anticode cardinality, and the (non)existence of linear diameter-perfect codes, i.e., lattice tilings by optimal anticodes -- and solve them in dimension $3$. We show that, for every integer $D\ge 0$, the largest cardinality of a diameter-$D$ subset of $A_3$ is $\binom{D+3}{3}+(D+1)\lfloor D^2/4\rfloor$, and this value is attained by the balanced difference of two discrete simplices. We then prove an integrality-refined simplex-packing obstruction: a sublattice of $\mathbb{Z}^n$ of asymmetric Manhattan distance greater than $D$ induces a lattice packing by $(D+1)Δ_n$ in $\mathbb{R}^n$. Combining this observation with the exact lattice-packing density of the tetrahedron yields a complete classification in dimension $3$: lattice diameter-perfect codes in $A_3$ exist precisely for $D=1$ and $D=2$. We also give the equivalent statement for perfect $B_h$ sets of cardinality four. Finally, we formulate a conjecture regarding optimal anticodes in arbitrary dimension, and restate it as an intersection problem for uniform multisets.


Lower bounds on the strength of the determinant

We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing $\mathrm{det}_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of $\mathrm{PGL}_n$ obstructs such a section.


The optimal constant for minimum weight feedback arc sets in oriented graphs

Let $D$ be an oriented graph (a digraph with no directed 2-cycles) with maximum degree $Δ\ge 1$, equipped with nonnegative arc weights of total weight $w(D)$, and let $\mathrm{fas}_w(D)$ denote the minimum weight of a feedback arc set of $D$. Alon (2002) proved $\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{1}{16\sqrt{2Δ}})w(D)$. We determine the optimal constant: \[\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{\sqrt{2}}{6\sqrtΔ})w(D).\] In fact, we show a stronger result: $\mathrm{fas}_w(D)\le\frac{1}{2}w(D)-\frac{\sqrt{2}}{12}\sum_v w_2(v)$, where $w_2(v)$ is the $\ell_2$-norm of the weights of the arcs incident with $v$. Both bounds are attained by the unit-weight directed triangle, so the constant $\sqrt{2}/6$ is best possible (already among unweighted oriented graphs). The proof combines the vertex-peeling scheme of Berger and Shor with a continuous random-ordering analysis: realizing the random order by independent uniform labels renders the expected local imbalance at each vertex exactly an integrated Khintchine-type functional, and the theorem reduces to the sharp evaluation \[\inf_{\|a\|_2=1}\int_0^1 \mathbb{E}|\sum_j a_j B_j(q)|\,dq = \frac{\sqrt{2}}{6},\] where the $B_j(q)$ are i.i.d. Bernoulli$(q)$ random variables, which we prove via Fourier analysis. The proof also yields a randomized, near-linear-time algorithm attaining the bounds in expectation.


New excluded minors for the class $\mathcal{M}_3$ of regular matroids

Engel, de Gaay Fortman, and Schreieder attach to each prime $\ell$ a minor-closed class $\mathcal{M}_\ell$ of regular matroids, whose excluded minors govern the failure of the integral Hodge conjecture for curve classes on very general principally polarized abelian varieties. For $\ell=2$ the class is the cographic matroids, with excluded minors $M(K_5)$ and $M(K_{3,3})$ by Tutte's theorem; for $\ell=3$ the only excluded minor explicitly identified so far is $M(K_{3,5})$, and the general characterization is [EGFS, Problem 8.8]. We exhibit five new excluded minors for $\mathcal{M}_3$, of ranks 8, 9, 9, 9, and 10; none contains $M(K_{3,5})$ or any of the other four as a minor, so the excluded-minor list of Problem 8.8 has at least six members, with excluded minors at every rank from 7 through 10. Combined with $M(K_7) \notin \mathcal{M}_3$, established by the same authors, and the minor-closedness of $\mathcal{M}_3$, some minor of $M(K_7)$ of rank between 4 and 6 is a further excluded minor, so the list has at least seven members, six of them explicitly identified. The list is structurally diverse: two bipartite apex constructions with automorphism group $S_4$, and three non-bipartite, non-planar rank-9 graphs with no such apex structure. Every verdict is certified by a finite $\mathbb{F}_3$-linear-algebra computation with explicit machine-checkable witnesses, cross-checked by independent implementations; the certificates further show that the five matroids lie outside the larger class $\widetilde{\mathcal{M}}_3$. We also record what are, to our knowledge, the first $\ell=5$ data on this corank-8 slice: $M(K_{3,5})$ and the rank-8 and rank-10 minors lie in $\mathcal{M}_5$ by explicit certificates, and the radical distance $d(M(K_{3,5}))=6$ is determined exactly.


Closed formulas for energy functions on tensor squares of higher-level perfect crystals in classical affine types

For every $l\geq1$, we construct explicit closed-form coordinate formulas for the local energy functions on the tensor products $B_l\otimes B_l$ of level-$l$ perfect crystals in classical affine types. A single finite-level max-linear formula covers all seven types: its two branches coincide in type $A_n^{(1)}$, yielding a cyclic maximum of partial sums, whereas in the remaining six types each branch is the maximum of finitely many explicit piecewise-linear expressions in barred coordinates, with type-dependent boundary data. We verify the defining local-energy recursion directly on the finite crystals and derive equivalent recursive forms, allowing the energy to be evaluated without applying the combinatorial $R$-matrix. Substitution into the KMN path character formula gives explicit positive coordinate path sums for the characters of all level-$l$ integrable highest weight modules. After principal specialization, the path exponent can be rewritten as a weighted sum of a position-independent adjacent-pair statistic; comparison with the specialized Weyl--Kac character formula yields a uniform family of level-$l$ Rogers--Ramanujan-type identities equating these sums with explicit infinite products. For a representative low-rank case at level two in each family, we display the complete adjacent-pair degree matrix and list the resulting identities.


Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes

Let $K$ be an $r$-dimensional simplicial complex. We prove that the spectrum of its $(r - 1)$-dimensional up-Laplacian is majorized by the conjugate degree sequence of its $(r - 1)$-dimensional faces: \[ {\mathbfλ}_{r-1}(K) \preccurlyeq {\mathbf d}_{r-1}^\top(K). \] We also establish a Brouwer-type inequality: for every integer $\ell \geq 1$, \[ \sum_{i = 1}^{\ell}λ_{r-1,i}(K) \leq \frac{r + 1}{2}f_r(K) + \frac{f_{r - 2}(K)}{r} \binom{\ell + 1}{2}, \] where $λ_{r-1,i}(K)$ denotes the $i$-th largest eigenvalue in the spectrum ${\mathbfλ}_{r-1}(K)$, and $f_t(K)$ denotes the number of $t$-dimensional faces of $K$. These results provide higher-dimensional analogs of the Grone-Merris-Bai theorem and the Brouwer-Kothari-Tudose theorem and recover the corresponding graph results when $r=1$.   We show that the Duval-Reiner conjecture on the majorization by the conjugate degree sequence of vertices fails in every dimension $r \geq 2$. More precisely, for every $n \geq r + 5$, we construct a pure $r$-dimensional complex on $n$ vertices that violates the conjectured inequality at the fifth partial sum.


Generalized quadrangles with a point-primitive and line-primitive automorphism group with socle $\PSp_4(q)$

Let $\SA$ be a finite thick generalized quadrangle, and let $G\leq \Aut(\SA)$ act primitively on both points and lines. Building on the almost simple reduction for point-primitive and line-primitive actions, we study the case where the socle of $G$ is the projective symplectic group $\PSp_4(q)$ with $q\ge 3$. We show that this hypothesis forces $\SA$ to be the classical symplectic quadrangle $W(3,q)$ up to duality.


Constructions for supersaturation of eventown problems

In this paper, we study the supersaturation problems of eventown. Given a family $\mathcal{A}$ of subsets of an $n$ element set, let op$(\mathcal{A})$ denote the number of distinct pairs $A,B\in \mathcal{A}$ for which $|A\cap B|$ is odd. We give extremal eventown constructions and show that for fixed $s\le2^{\lfloor \frac{n}{2} \rfloor}-2$, there exists a collection of $2^{\lfloor\frac{n}{2}\rfloor}+s$ even-sized subsets of an $n$ element set that contains exactly $s\cdot 2^{\lfloor \frac{n}{2} \rfloor-1}$ pairwise intersections of odd size. This extends the range of $s$ in a conjecture proposed by O'Neill from $2^{\lfloor \frac{n}{2} \rfloor}-2^{\lfloor \frac{n}{4} \rfloor}$ to $2^{\lfloor \frac{n}{2} \rfloor}-2$. We also give a construction using symmetric designs to prove that when $k$ is even and $4k-1$ is a prime power, there exists a collection of $2^{\lfloor\frac{4k-1}{2}\rfloor}+s$ even-sized subsets of a $4k-1$ element set $\mathcal{A}_s$ with $op(\mathcal{A}_s)=s \cdot 2^{{\lfloor\frac{4k-1}{2}\rfloor}-1}$, $1\leq s\leq4k-1$.


Flag Space, Matroidal Schur Algebras and the Steinberg Representation

We define via generators and relations an extended version of the matroidal Schur algebras introduced in earlier work of the author with Tom Braden. In the case of projective geometries, we relate the representation theory of the resulting algebras to the structure of the modular Steinberg representation of $GL(n,q)$. In the general case we show that the extended matroidal Schur algebras have a triangular, or Reedy, decomposition and are thus quasi-hereditary. We also obtain a new description of the original matroidal Schur algebras and clarify the connection to the flag spaces of Brylawski--Varchenko.


Multiplier obstructions for Legendre pairs of length 333

A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture. We study the structured case in which both sequences are fixed by a common subgroup $H\leq(\mathbb Z/333\mathbb Z)^\times$ acting by coordinate multiplication. We prove that such a pair can exist only when $|H|\leq 6$. After a mod-3 compression reduces the problem to an order-108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order-9 subgroup is eliminated analytically: its orbit structure restricts the 9-compressed entries to $\{\pm1,\pm17,\pm19,\pm35,\pm37\}$; the Legendre equations force a $+17,-17$ pair in one compressed sequence, and a single-shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full-image compression, a row-sum congruence, exact meet-in-the-middle enumeration, and proof-carrying pseudo-Boolean encodings. The solver-assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common-multiplier symmetry only; the unrestricted existence problems remain open.


Infinite Sidon-type sets for zero-sum linear forms

Let $h \geq 2$, and let $\mathbf{b} = (b_1,\dots,b_h)\in \mathbb{Z}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$, let $r_{A,\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\ldots,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1x_1+\cdots+b_hx_h=n$. We study density restrictions on sets $A$ for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets.   In the case $\mathbf{b} = (c_1,-c_1,\dots,c_k,-c_k)$, we prove that if $A(x)/(x/\log x)^{1/2k}\to\infty$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $A(x)\gg x^{1/2k}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$. This recovers Chen's theorem on $B_{2k}$-sequences. For general zero-sum vectors $\mathbf{b}$, we prove analogous bounds under gap conditions: if $a_{n+1}-a_n=o(n^{h-1}\log n)$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $a_{n+1}-a_n\ll n^{h-1}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$.


An Improved Upper Bound for Colorings Without Symmetrically Colored $k$-Term Arithmetic Progressions

Given a coloring $c$ and an even $k\ge 4$, a nontrivial $k$-term arithmetic progression~($k$-AP) $a,a+d,\ldots,a+(k-1)d$ is called symmetrically colored if $c(a+(i-1)d)=c(a+(k-i)d)$, $\forall i\in[k/2]$. Deng, Tidor, and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no such 4-APs, and gave an $O(N^{\log_{22}3})$-coloring of $[N]$. We give an $O_k(p)$-coloring of $\mathbb Z/p^{k^2/4}\mathbb Z$ without such $k$-APs for every even $k\ge 4$ and every prime $p>k$, and hence an $O(N^{4/k^2})$-coloring of $[N]$, improving the exponent in the upper bound for $4$-APs from $\log_{22}3$ to $1/4$. The construction combines a carry-control coloring of base-$p$ digits with a layered field norm mapping. Together with Behrend-style product colorings, our result for $4$-APs gives $h(N)\leq N^{1/4+o(1)}$ in Erdős's Problem~160 on coloring every nontrivial 4-AP with at least three colors. This result also yields $ρ_4(α)=O_\varepsilon(α^{5-\varepsilon})$ for every $\varepsilon>0$, improving the bound toward Ruzsa's question. Our result for $k$-APs disproves Gowers' conjectured lower bound for all even $k\ge6$ for the first time.


Pattern avoidance in permutations and their rotations

A rotation of a permutation is a new permutation obtained by moving the first several terms of the permutation to the end of the permutation. A circular permutation is the set of all rotations of a permutation. The enumerations of permutations and circular permutations avoiding patterns of length three and four are well studied. In this paper, we provide exact formulas for the number of permutations whose first $k\geq 2$ rotations all avoid a given pattern of length three, as well as the number of permutations whose first three rotations respectively avoid the rotations of a given pattern of length three. In contrast to permutations and circular permutations avoiding patterns of length three, the Wilf-equivalence classes under study are entirely determined by complements and reverses. We also classify and enumerate permutations whose first two rotations avoid different patterns of length three.


Application of the Combinatorial Nullstellensatz to magic-type graph labelings

Let $G=(V,E)$ be a simple graph, and let $k\geq 2$ be an integer. For an edge labeling $h:E(G)\to \mathbb{Z}_{k} \backslash \{0\}$, define the induced vertex label by \[ h^+(v)=\sum_{e \ni v} h(e) \pmod{k}. \] For $t\in \mathbb Z_k$, we say that $G$ is \emph{$t$-sum $\mathbb Z_k$-magic} if there exists such a labeling $h$ satisfying \[ h^+(v)=t \qquad\text{for all }v\in V. \] We say that $G$ is \emph{$\mathbb Z_k$-magic} if $G$ is $t$-sum $\mathbb Z_k$-magic for some $t\in \mathbb Z_k$. Similarly, if there exists an edge labeling $h: E(G) \to \mathbb{Z}_{k} \backslash \{0\}$ such that the induced vertex labeling $h^+(v)=\sum_{e\ni v} h(e)$ (mod $k$) is injective, then $G$ is called \emph{$\mathbb{Z}_{k}$-antimagic}. In this paper, we use the Combinatorial Nullstellensatz to analyze these two types of magic graph labelings.


Shellability of relative squeezed balls and spheres

Squeezed balls and spheres, introduced by Kalai, form a rich class of triangulated complexes arising from subcomplexes of cyclic polytopes, with well-understood shellability properties. Recently, Novik and Zheng introduced relative squeezed balls, obtained as differences of squeezed balls, and used them to construct large families of highly neighborly simplicial spheres. While these complexes are known to be constructible, their shellability has remained open. In this paper, we resolve this question by proving that both relative squeezed balls and their boundary complexes are shellable. We provide explicit shelling orders and characterize restriction faces, thereby establishing strong combinatorial structure for this new class of complexes.


Combinatorial formula for the Moore-Penrose inverse of the complex signless Laplacian of an oriented graph

We find necessary and sufficient conditions for the rank of the signless incidence matrix of a weakly connected oriented graph with non-zero complex edge weights. We use this to find the combinatorial formulas for the Moore-Penrose inverse of the complex signless incidence and complex signless Laplacian matrix of a weakly connected oriented graph with non-zero complex edge weights. This resolves the open problem posed in the concluding remarks of Barik et. al. (Discrete Mathematics 349 (9), 115117, 2026).


Comments on "Condensation of the digraph associated with a reciprocal matrix and a vector", arXiv:2607.10279

Since manuscript \cite{R}, recently posted on arXiv, is closely related to our previous work, we would like to clarify some facts about it. Namely, we would like to point out that several results in \cite{R} are already known (Theorem 1 and Corollaries 2, 3 and 4) and some observations regarding our work \cite{FJ} are not correct. We also notice that Rédei's Theorem, a classical result in graph theory, gives a simple tool to show Theorems 5 and 9.


The sharp exponent for the minimal distance problem

We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem.


A 13-vertex counterexample to $e$-log-concavity for chromatic quasisymmetric functions

We exhibit a connected natural unit interval graph on $13$ vertices whose chromatic quasisymmetric function has an elementary-basis coefficient that is positive, palindromic, and unimodal, but not log-concave. For the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $λ=(6,5,1,1)$, the coefficients of $q^5,q^6,q^7$ in $[e_λ]X_{G(h)}(\mathbf{x};q)$ are $1,6,38$; hence $6^2<1\cdot38$. This disproves Conjecture 5.3 of Abreu--Nigro and its later formulations by Tom and by Sagan--Tom. The calculation is certified by a standalone deterministic verifier using exact integer and rational arithmetic.


The inverse problem for the Steiner-Wiener index of trees

For a connected graph $G$ and a set $S\subset V(G)$, the Steiner distance $d_G(S)$ is the minimum number of edges in a connected subgraph of $G$ containing $S$. The Steiner-Wiener $k$ index is defined by $\mathrm{SW}_k(G) = \sum_{S\subset V(G), |S|=k} d_G(S)$. We study the inverse problem for this invariant restricted to trees: for fixed $k$, which positive integers occur as $\mathrm{SW}_k(T)$ for a finite tree $T$? We prove that all sufficiently large positive integers occur as $\mathrm{SW}_k(T)$ for some finite tree $T$ if and only if $k$ is even. For odd $k$, we further show that the set of attainable values has asymptotic density of order $k^{-δ}(\log k)^{-3/2}$, where $δ$ is the Erdős-Tenenbaum-Ford constant.


The Erdős-Lovász Tihany Conjecture holds for all even-hole-free graphs

Let $s, t\ge2 $ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $χ(G[S ]) \ge s$ and $χ(G[T ]) \ge t$. The Erdős-Lovász Tihany Conjecture from 1968 asserts that every graph $G$ satisfying $ω(G)<χ(G)=s+t-1$ is $(s,t)$-splittable. A vertex of a graph is bisimplicial if the set of its neighbors can be expressed as the union of two cliques. Let $G$ be a graph with $ω(G)<χ(G)=s+t-1$. We prove that if $G$ does not contain $C_4$ as an induced subgraph and every induced subgraph of $G$ has a bisimplicial vertex, then $G$ is $(s,t)$-splittable. Combining our result with a recent result of Chudnovsky and Seymour, which states that every non-empty even-hole-free graph has a bisimplicial vertex, we obtain that the Erdős-Lovász Tihany Conjecture holds for all even-hole-free graphs.


On (3,1)-regular graphs with one more vertex than edges

Sequence A339987 of the OEIS counts (3,1)-regular graphs having one more vertex than edges by half the number of vertices. A recurrence relation satisfied by this sequence was guessed by Kauers and Koutschan in 2023. We confirm it in three ways: first, by a representation as the diagonal of a triple sum and an elaborate variant of traditional creative telescoping that makes an a posteriori validation possible; second, by a residue representation and a direct calculation by reduction-based creative telescoping; third, by a combinatorial recurrence on graph families and a calculation by differential elimination. Each of those three approaches leads to a formally complete proof and involves a computer calculation in one way or another.


Triple arrays from ovals in finite projective planes

In this paper we prove that whenever a projective plane of odd order n contains an oval, it may be used to construct a triple array with n + 1 rows, n.n columns and n(n + 1) symbols. In particular, for any odd prime power s, we use the projective plane over the field of order s to construct an explicitly given (s + 1) times s.s triple array on s(s + 1) symbols. This is only the third infinite series of triple arrays (up to transposition) to be found. Our construction proves a recent conjecture of Gordeev and Ohman (2025) in the case of odd prime powers.


Generic properties of discrete Steklov eigenfunctions

Let $G=(V,E)$ be a finite connected graph with boundary $B$. We prove that for a generic positive edge weight function $w \in \mathbb{R}^{|E|}$, the Steklov eigenvalues of $(G,B,w)$ are simple and every Steklov eigenfunction does not vanish on the boundary. More precisely, the exceptional weights are contained in a zero set of a non-identically zero polynomial and hence form a set of Lebesgue measure zero and Hausdorff dimension at most $|E|-1$. Our results provide a discrete extension of the genericity theorem for the Steklov problem on compact manifolds.


A Matrix-Degree Obstruction to Rational Generation of Boolean-Lattice Pseudo-Roots

For the neighborhood seed associated with the four-vertex path $P_4$, the diamond operations do not recover all Boolean-lattice pseudo-roots. The corresponding question for unrestricted rational operations in the free skew field is subtler: the seed map has an invertible linearization and therefore a unique formal inverse near every generic scalar point. We prove that this formal inverse is not free rational. A symmetric one-parameter curve of $2 \times 2$ matrix outputs has a formal inverse whose coefficient field contains an element of degree three over $\mathbb{Q}(t)$. An exact elimination in a quadratic Pauli algebra produces the irreducible cubic. Its conjugate inverse branches are unramified, forcing the generic matrix degree of the seed map to be at least three in every size $n \ge 2$. This contradicts the degree-one consequence of any free rational inverse. The same matrix-degree argument, without specializing a hypothetical inverse, extends the obstruction to every graph containing an induced $P_4$.


Graphon as a Bridge between Graphs and Manifolds

We show that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs. Specifically, the graph-to-manifold approximation used in manifold learning can be regarded as the composition of a graph-to-graphon convergence and a graphon-to-manifold convergence in a certain sense. Furthermore, we establish a monotonicity inequality which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons. Using this inequality, we find relations among conductance, maxcut problem, capacity, and packing radius, as well as their limiting behaviours under graph-to-graphon and graphon-to-manifold convergences; some of these relations are novel even for simple graphs and closed manifolds.


Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs

Let $G$ be a connected simple vertex-transitive graph on $n$ vertices with degree $d$, and let $G_p$ be the random spanning subgraph obtained by retaining each edge independently with probability $p$. Motivated by a conjecture of Bedert, Draganić, Müyesser, and Pavez-Signé on Hamilton cycles in percolated Cayley graphs, we establish its matching and $2$-factor consequences for the larger class of all connected vertex-transitive host graphs. First, for every $A>0$, there is $C=C(A)>0$ such that, for every $N\ge n$, the condition $(1-p)^d\le N^{-C}$ implies, with probability at least $1-N^{-A}$, that $G_p$ has a perfect matching when $n$ is even and that $G_p-v$ has a perfect matching for every vertex $v$ when $n$ is odd. Second, if $ν_2(H)$ is the maximum number of pairwise edge-disjoint spanning $2$-factors in $H$, then, for every $A>0$ and $0<ε<1$, \[   ε^2pd\ge64(A+6)\log(2n) \] implies \[   \mathbb P\left(   ν_2(G_p)\ge\left\lfloor(1-ε)\frac{pd}{2}\right\rfloor   \right)\ge1-n^{-A}. \] Consequently, $pd\ge C\log n$ guarantees both the appropriate matching property and a spanning $2$-factor with high probability, uniformly over all connected vertex-transitive graphs. If $pd/\log n\to\infty$, then $ν_2(G_p)=(1+o(1))pd/2$ with high probability. The coefficient $1/2$ is best possible because each spanning $2$-factor contains $n$ edges, whereas $G_p$ contains about $pnd/2$ edges.


Induced Subgraph Bounds on the Zero Forcing Number and a $(χ, ω, Z)$-Conjecture

Let $G$ be a graph with chromatic number $χ(G)$, clique number $ω(G)$ and zero forcing number $Z(G)$. We establish new lower bounds on $Z(G)$ in terms of induced triangle-free subgraphs. In particular, we show that if a graph $G$ contains an induced triangle-free subgraph $H$ with minimum degree $δ(H) \ge 3$, then $Z(G)\geδ(H)+1$. Motivated by this result and the bound $χ(G) \leq Z(G) + 1 $ by Taklimi (2013), we conjecture that   \begin{equation*}   χ(G) \leq \left \lceil \frac{ω(G)+Z(G)+1}{2}\right\rceil. \end{equation*} As supporting evidence, we prove that the conjecture holds for triangle-free regular graphs and also provide numerical evidence.


Generalised flip order on the faces of nestohedra

Classical shuffle products on permutations and binary planar rooted trees (i.e., on the vertices of permutohedra and associahedra) admit descriptions in terms of intervals in the weak Bruhat order and the Tamari order, respectively. Palacios and Ronco extended these products as well as their interval description to surjections and planar rooted trees (i.e. on all faces of permutohedra and associahedra). In this article, we present a broad generalisation of this phenomenon. We show that the shuffle product on faces of certain families of nestohedra admits an interval description with respect to the generalised flip order, a partial order defined on the faces of nestohedra through elementary splitting and fusion operations on the tree-like combinatorial objects encoding them. The generalised flip order extends the flip order of Barnard and McConville from vertices to all faces of nesthedra. We further compare it with the facial weak order of Dermenjian-Hohlweg-Pilaud and the generalised Tamari order of Ronco, and we provide its characterisation in terms of (generalised) inversions.


Combinatorial geometry of the 2D Toda lattice and Davey Stewartson equation

The KP equation is a prototypical $(2+1)$-dimensional integrable PDE. Its soliton solutions are famously parametrized by the Sato Grassmannian. In seminal work, Kodama and Williams made the surprising discovery that the combinatorics of soliton solutions are intimately related to the combinatorics of the totally positive Grassmannian as pioneered by Postnikov. They introduced novel algorithmic methods inspired by polyhedral structures arising from tropical geometry. Soliton solutions to the 2D Toda lattice and the Davey--Stewartson equation, two closely related integrable systems with soliton solutions, are also classified by the Sato Grassmannian. Kodama suggested that the methods of his work with Williams could generalize to these two integrable equations. In this work, we show that this is indeed the case. We derive algorithms to produce contour plots from elements in the totally nonnegative Grassmannian in both cases. In the asymptotic setting, we recover and refine previous work of Biondini and Wang; as well as Biondini, Kireyev and Maruno.


The Duval--Reiner Conjecture: Counterexamples and the Second Partial-Sum Inequality

Let \(F\subseteq\binom{V}{q}\) be a \(q\)-uniform family on a finite vertex set \(V\). Write \(s_r(F)\) for the sum of the \(r\) largest eigenvalues of its simplicial up-Laplacian and \(d_F(v)\) for the degree of \(v\in V\). Then $D_r(F)=\sum_{v\in V}\min\{d_F(v),r\}$ is the \(r\)-th partial sum of the conjugate degree sequence of \(F\). The majorization assertion in the Duval--Reiner conjecture [Trans. Amer. Math. Soc., 2002] states that \(s_r(F)\le D_r(F)\) for every \(q\)-uniform family \(F\) and every \(r\ge1\). We disprove this assertion in two complementary senses: for every \(r\ge5\), there is a strict counterexample at index \(r\) in some uniformity, while every uniformity \(q\ge3\) admits a strict counterexample at some index \(r\ge5\). In contrast, we prove the universal inequality \(s_2(F)\le D_2(F)\) and classify all equality cases. The counterexamples are obtained from two \(3\)-uniform seeds with explicitly computed characteristic polynomials through defect-preserving ridge-whiskering and set-complement duality. For the second partial sum, core completion reduces the problem to the boundary matrix of a complete simplex, where Ky Fan variational and compression arguments yield both the inequality and its equality classification.


Restricted $r$-Stirling numbers of the second kind

In this paper, we introduce the $(r,d)$-Stirling numbers of the second kind, a new generalization of the $r$-Stirling numbers obtained by imposing a distance restriction: for any two elements $i$ and $j$ within the same subset, we require that $|i-j|>d$. We establish recurrence relations, an explicit formula, ordinary and exponential generating functions, a reduction identity connecting these numbers to the standard $r$-Stirling numbers, and several combinatorial identities.


How Close is a Tree to a Euclidean Minimum Spanning Tree?

Let $Γ$ be a straight-line crossing-free drawing of a tree $T$. A \emph{bad pair} in $Γ$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $Γ$ is smaller than the length of the longest edge in the path connecting them in~$Γ$. When $Γ$ has no bad pairs, $Γ$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $Δ$, we prove the $Δ^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.


Exact certification of the coordinate fields of the triangle-free Exoo-Ismailescu unit-distance graphs EI17 and EI19 (HoG 51375, 51376): a solvable-non-solvable dichotomy (origami vs. S20) and the Laman-number conjecture

We certify, exactly, the coordinate fields of a faithful planar realization of two neighbouring triangle-free Exoo-Ismailescu unit-distance graphs (UDGs), and show they realize the two opposite extremes of the constructibility hierarchy. The 17-vertex graph EI17 (House of Graphs 51375) is the smallest triangle-free UDG with chromatic number 4; the 19-vertex graph EI19 (HoG 51376) is its state-of-the-art origami neighbour. In both, fixing a rational base edge, the remaining vertices are intersections of unit circles -- each on the radical axis of its two neighbours, a tower of square roots over the free angles -- and a small closure system locks the realization. For EI19 the base lies in Q(sqrt 2, sqrt 5, sqrt 7) and a single free angle has an irreducible degree 12 = 2^2*3 minimal polynomial with Galois group the solvable transitive group 12T236 (order 2304 = 2^8*3^2): not ruler-and-compass, but origami-constructible (the cubic Beloch fold O6 necessary, in casus irreducibilis). For EI17 two free angles are locked by two closures, whose resultant is irreducible of degree 20 = 2^2*5 with Galois group the full symmetric group S20 (a Frobenius census exhibits a 17-cycle, forcing A20 by Jordan, and an odd 20-cycle, raising it to S20): non-solvable, so the coordinates are not expressible by radicals -- neither compass nor origami of any fold order. Thus the smallest triangle-free 4-chromatic UDG is the generic, maximally exotic case, the exact opposite of its origami neighbour. We give the full certification pipeline as explicit algorithms, record two methodological pitfalls, and read the pair through a conjectural bridge between the Laman number and the Galois group.


On the possible cardinalities of quantum Latin squares

In this paper, we show that for any integer \(v \ge 8\) with \(v \ne 9,11,23\), a quantum Latin square of order $v$ exists for every cardinality \(c \in [v,\,v^{2}] \setminus \{\,v+1\,\}\).


The Fibonacci Rectangle Game: Two First-Move Classes and a Triangle-Induced Choice

A square-adjoining rectangle game generates the Fibonacci numbers and the Fibonacci spiral from a simple geometric rule. If one starts from a square, the four possible first moves are all equivalent by rotation. If one starts instead from a non-square rectangle, there are still four geometric placements for the first square, but they split into exactly two equivalence classes: long-side-first and short-side-first. We show that both classes are governed by the same Fibonacci-type recursion with different initial conditions, and that in both cases the successive aspect ratios converge to the golden ratio (phi). We then add a brief geometric remark: the Hypotenuse-Axis Intercept (HAI) construction from a right triangle produces a natural ordered seed whose outward and inward branches determine precisely those two first-move classes.


The chromatic number of Euclidean space with dense color classes

In this note we construct colorings of Euclidean space $\mathbb{R}^n$ with finitely many colors such that any two points at unit distance have different colors and, in addition, each color class is dense in $\mathbb{R}^n$. In particular, 12 dense colors suffice to color $\mathbb{R}^2$. In arbitrary dimension, we show that $nχ(\mathbb{R}^n)+1$ colors suffice, where $χ(\mathbb{R}^n)$ denotes the chromatic number of $\mathbb{R}^n$ in the standard formulation.


On Boolean sublattices of finite partition lattices

We investigate maximal Boolean sublattices of the partition lattice Part(U) of a finite universe U. First, any largest size Boolean sublattice of Part(U) can be formed using all partitions whose blocks are subtrees of a tree with vertex set U. It is shown that a maximal Boolean sublattice of Part(U) always contains the least and the largest elements of Part(U). Boolean sublattices B of PartU containing 0 are characterized by a certain condition imposed on the cycles of a linear hypergraph corresponding to the atoms of B on the set U. We show that B can be extended to a Boolean sublattice of Part(U) with a largest size, if and only if the hypergraph induced by its atoms is a hypertree. This is the case when B is formed by all the partitions whose blocks are intervals in a generalized sense. The main result states that all partition lattices of height at least three have maximal Boolean sublattices for all possible dimensions at least three


Higher cluster tilting objects in locally finite triangulated categories

We study higher cluster tilting objects through covering functors from derived categories of hereditary algebras. The covering formalism reduces the existence problem to the equivariant problem of finding \(G\)-stable \(d\)-cluster tilting objects in \(d\)-cluster categories. For triangulated categories with finitely many indecomposable objects this gives a complete ADE existence criterion and explicit counting formulas. We also prove that, in finite Frobenius models, the endomorphism algebras of all \(d\)-cluster tilting objects are derived equivalent. Applications are given to Cohen--Macaulay finite categories, finite noncommutative crepant resolutions, and rigidity dimensions of representation-finite self-injective algebras.


Hamiltonian cycles in 7-tough $(P_4\cup P_1)$-free graphs

Shan~[J. Graph Theory (2026)] proved that every 23-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. We improve this bound to 7 by replacing the final cut analysis in Shan's framework with an asymmetric separation criterion and a cograph covering lemma.


Negative Latin-Square-Type Partial Difference Sets in Non-Elementary Abelian 2-Groups

Using cubic cyclotomic classes, character theory, and product constructions motivated by generalized Denniston partial difference sets, we construct negative Latin-square-type partial difference sets in $\mathbb{F}_{2^6}^{+}\times\mathbb{Z}_4^4$, $\mathbb{Z}_4^4\times\mathbb{F}_{2^{10}}^{+}$, $\mathbb{F}_{2^{10}}^{+}\times\mathbb{Z}_8^4$, $\mathbb{Z}_8^4\times\mathbb{F}_{2^{14}}^{+}$, and $\mathbb{Z}_4^4\times\mathbb{Z}_{16}^2$. The first four constructions replace cubic cyclotomic partitions by partitions of non-elementary abelian 2-groups having the same character-value patterns. To the best of our knowledge, these are the first partial difference sets with the stated parameters in the indicated groups.


Polyhedral Maps of Cubic Graphs with given Automorphism Groups

L. Babai introduced a method for constructing a cubic graph whose automorphism group is isomorphic to a given finite group $G$, obtained by modifying a corresponding Cayley graph of $G$. Building on this approach, we construct a cubic graph that admits a polyhedral map whose automorphism group, as well as the automorphism group of the polyhedral map itself, is isomorphic to $G$.


For a graph $G$ of order $n$, with adjacency eigenvalues $λ_1(G) \geq \cdots \geq λ_n(G)$, the \emph{energy} of $G$ is defined to be \[\mathcal{E}(G)=\sum_{i=1}^{n} |λ_i(G)|.\] A well-known conjecture from the 1980s by Fajtlowicz states that for any graph $G$, \[\mathcal{E}(G) \ge 2\left(n-α(G)\right),\] where $α(G)$ denotes the independence number. We prove this conjecture.


A direct injection for the strong $q$-log-convexity of Touchard polynomials

We provide a direct injection for the well-known strong log-convexity of the Bell numbers $B_n$, that is $B_mB_n\le B_{m-1}B_{n+1}$ for every $1\le m\le n$. Our injection $Π_m\timesΠ_n\to Π_{m-1}\timesΠ_{n+1}$, where $Π_n$ denotes the set of all partitions of $[n]$, preserves the total number of blocks in the pair of partitions. In other words, it is also an injection for the strong $q$-log-convexity of Touchard polynomials, a result established by Chen, Wang, and Yang using analytical arguments. As an application of the injection, we also recover a related result of Chern, Diaconis, Kane, and Rhoades.


A Sharp Ramsey Theorem for Admissible Colorings of Ordered Cliques

Let \(f(k)\) be the minimum integer \(N\) such that any red--blue edge-coloring of the ordered complete graph on \(N\) vertices contains a set of \(k\) vertices whose induced coloring is admissible. In this note, we obtain the exact value of $f(k)$ for $k\ge 3$, which confirms a conjecture posed by Bradač, Liu, Wu and Xu.


Classification of two-distance-transitive Cayley graphs of the semi-dihedral groups

The class of 2-distance-transitive graphs naturally generalizes distance-transitive graphs and plays a central role in algebraic graph theory. Classifying such graphs for a prescribed underlying group is a key open problem. A vertex-transitive graph $Γ$ is said to be $2$-distance-transitive if, for each $i\in \{1,2\}$, any two pairs of vertices with identical distance $i$ in $Γ$ can be mapped to each other via some automorphism of the graph. In this paper, we present a complete classification of all $2$-distance-transitive Cayley graphs of the semi-dihedral groups.


Hitting time mixing for random $k$-cycles

In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$. For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$. Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions. The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework. We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.


Negative-index $u$-deformed homogeneous functions and Ramanujan partial theta functions

We introduce and study a class of \(u\)-deformed homogeneous functions of complex order \(\rr_α(x,y;u\,|\,q)\), obtained by extending \(u\)-deformed homogeneous polynomials beyond the nonnegative integer regime. We establish their basic analytic and algebraic properties, including convergence criteria, recurrence relations, \(q\)-derivative formulas, and an operator-theoretic realization through a deformed \(q\)-exponential operator. A central feature of the theory appears in the negative-index sector: for every nonnegative integer \(n\), the functions \(\rr_{\minus n\minus1}(x,y;u\,|\,q)\) admit finite triangular decompositions in terms of shifted Ramanujan partial theta functions. We also prove the inverse triangular transformation, showing that the corresponding partial theta functions can be recovered from the negative-index \(u\)-deformed homogeneous functions. We also derive pantograph-type \(q\)-difference equations for these functions. Finally, we derive specialized forms of the negative-index theory in Cauchy and Stieltjes--Wigert-type regimes, including a boundary geometric case and genuine partial-theta regimes.


Given a graph $H$, the extremal number $ex(n,H)$ is the maximum number of edges in an $n$-vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjecture of Erdős and Simonovits states that for any rational $γ\in (1,2)$ there exists a single bipartite graph $H$ satisfying $ex(n,H)=Θ(n^γ)$. Among other results, the conjecture has been verified for all $γ=1+a/b$, where $b>a^2$, by Jiang and Qiu and for all $γ=2-a/b$, where $b>\max\{a, (a-1)^2\}$, by Conlon and Janzer.   In this paper, we establish the rational exponents conjecture for many $γ$ near the center of the interval, namely, for all $γ=1+\frac{rt-1}{2rt+2r}$, where $r,t$ are natural numbers satisfying $t\geq 2$, $r\geq 2t+3$.


Maximum Likelihood Estimation on the Grassmannian of Lines

We study the positive Grassmannian through the lens of algebraic statistics. A closed formula is presented for the maximum likelihood degree of the Grassmannian of lines. We conjecture that the probability simplex contains a unique local maximum, and we present computational evidence for this.


Bounds and Limitations on Codes Achieving List Recovery Capacity

In coding theory, list recoverability is a fundamental concept which robustly captures how ``spread-out'' codewords are in a code. More formally, given a code $C \subseteq Σ^n$ and input lists $S_1, \dots, S_n \subseteq Σ$ of size at most $\ell$, list recoverability requires that there are at most $L$ codewords $c \in C$ such that $c_i \in S_i$ for at least $(1-ρ)n$ choices of $i \in [n]$. List recovery is an important question which has found applications in many areas, including complexity theory, property testing, compressed sensing, streaming algorithms, and cryptography.   As our first main result, we establish a tight ``generalized singleton bound''. Formally, we show that for constant $\ell, L,ρ$ and sufficiently large alphabets $Σ$, if we define $R^*=\frac{L+1-\ell}{L}-\frac{L+1}{L}ρ$, it is possible for a $(ρ,\ell,L)$ list-recoverable code to have rate $R^*-ε$ but impossible to have rate $R^*+ε$. One direction of our result already directly generalizes and improves a weaker impossibility result due to Goldberg, Shangguan, and Tamo.   For our second main result, we prove that there is a fundamental shortcoming in existing methods that aim to construct explicit, optimal list-recoverable codes. Indeed, recent work has constructed explicit codes achieving list-decoding capacity (along with other related properties) using a framework introduced in the work of Alon--Edmonds--Luby (AEL). We give a meta-analysis of such constructions by presenting an ``AEL framework'' which captures all such recent constructions in the literature. Within this framework, we show that no AEL-based code can break a recently-identified list-recovery barrier for additive and linear codes.


On the Ramsey number of a graph obtained by attaching a pendant to a path with length 1 modulo 4

In this paper, for odd $n$, we consider the graph $T_n$ with vertex set $\{x_1,....,x_n,x_{n+1}\}$ and edge set $\{x_{1}x_{2},x_{2}x_{3},...,x_{n-1}x_{n}\} \cup \{x_{\frac{n+1}{2}}x_{n+1}\}$ and prove that the Ramsey number $R(T_n,T_n)$ is equal to $\frac{3n+1}{2}$ for $n \in [1]_{4}$.


New lower bounds for binary constant-weight codes: $A(23,6,10)\geq 2979$ and $A(24,6,10)\geq 4214$

Let $A(n,d,w)$ denote the maximum size of a binary constant-weight code of length $n$, minimum distance $d$, and weight $w$. We construct explicit codes proving $A(23,6,10)\ge 2979$ and $A(24,6,10)\ge 4214$. These improve the best surviving explicit codes of sizes 2969 and 4174 and surpass the corresponding 1990 bounds 2970 and 4200 of Brouwer, Shearer, Sloane and Smith, whose code listings were lost. We also obtain $A(23,6,11)\ge 3539$ and $A(24,6,8)\ge 1855$. All four bounds are now listed in Brouwer's online table. The constructions use a coordinate decomposition in which one half is fixed to a known code and the complementary half is selected from its full cross-compatible pool using CHILS for maximum-weight independent set. For the 2969-word $A(23,6,10)$ incumbent, exact computations with two solver families prove insertion maximality and exclude every improving exchange deleting at most three codewords. We also analyze codes invariant under prime-order permutations: several cycle types are excluded exactly, the $5+1^{18}$ type has upper bound 499, and reproducible heuristic saturation evidence is reported for the remaining types, with $13+1^{10}$ left open. Code files, an independent validator, model descriptions, and computational logs are released.


Perfect state transfer in Grover walks on normal Cayley graphs

A Cayley graph $\operatorname{Cay}(Γ,S)$ over a finite group $Γ$ is said to be normal if its connection set $S$ is a union of some conjugacy classes of $Γ$. This paper investigates perfect state transfer in Grover walks on normal Cayley graphs. The Grover walk is a widely studied discrete-time quantum walk. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on normal Cayley graphs. As applications, we derive explicit spectral criteria for perfect state transfer on Cayley graphs over abelian groups, dicyclic groups, and dihedral groups. These results yield several infinite families of Cayley graphs exhibiting perfect state transfer. We further obtain simple combinatorial characterizations of the existence of perfect state transfer on Cayley graphs over dihedral and dicyclic groups. Our general characterization also recovers a number of previously known results as special cases. As a further consequence, we obtain a complete characterization of perfect state transfer on unitary Cayley graphs. In particular, we prove that exactly four graphs in the class of unitary Cayley graphs exhibit perfect state transfer.


Resolution of the ENO-TV conjecture: a parity dichotomy

We resolve the ENO--TV conjecture, a discrete coercivity problem in compactness theory for entropy-stable approximations of hyperbolic conservation laws. For order-$k$ essentially non-oscillatory (ENO) reconstruction from compactly supported cell averages, it asks whether the nonnegative ENO source times the $(k-1)$st power of the amplitude uniformly controls the $(k+1)$st absolute-jump moment. We prove a parity dichotomy: the estimate holds for odd $k\ge3$ and fails for even $k\ge4$; the known second-order case completes the classification. Localization gives a selection-independent finite-difference functional uniformly comparable to the source and reduces the conjecture to discrete interpolation. For odd orders, summation by parts reveals a hidden square; a discrete Gagliardo--Nirenberg inequality yields coercivity. For even orders, Euler-polynomial blocks from the functional's polynomial kernel yield counterexamples that persist under arbitrarily small perturbations making all affected ENO comparisons strict. We also prove two coercive estimates for every $k\ge2$: control of jumps larger than a fixed fraction of the amplitude and of local blocks modulo sampled polynomials of degree at most $k-2$. Via the Cayley--Sylvester decomposition, we compute the dimensions of homogeneous first-cohomology spaces for the lattice shift on polynomial jump profiles. At fourth order, for a cubic flux and a globally strictly convex entropy, a total-degree-seven component of a reduced entropy-flux mismatch represents a nonzero class on profiles of degree at most two and hence has no translation-invariant finite-stencil $C^7$ local primitive at the zero constant state. Odd-order coercivity persists on globally quasi-uniform meshes, whereas for each $k\ge2$ it fails on a fixed irregular mesh even though every interface contribution remains nonnegative. This failure is due to the mesh geometry.


A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture

We prove Norine's conjecture: every red--blue edge-coloring of the \(n\)-dimensional hypercube \(Q_n\), \(n\geq2\), in which antipodal edges have opposite colors contains a monochromatic path joining some vertex to its antipode. From a hypothetical counterexample we construct an antipodally equivariant, augmentation-preserving chain map from the cellular chains of the cubical boundary of a cube to subdivision-invariant polyhedral chains on a sphere of one lower dimension. A purely algebraic chain-level Borsuk--Ulam obstruction rules out this map.


A complete solution to the Boots-Royle/Cao-Vince conjecture

Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge and a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for $n\geq 9$. Tait and Tobin (JCTB, 2017) proved the conjecture for sufficiently large order. In this paper, we completely resolved the Boots-Royle/Cao-Vince conjecture.


Geometries admitting trialities for the symmetric and alternating groups

In incidence geometry, a triality is a symmetry cyclically exchanging triples of types of elements. Requiring geometries with trialities to satisfy standard regularity conditions makes their construction highly non trivial, and known examples are rare. In this paper, we present two infinite families of flag transitive, thin and residually connected geometries admitting trialities and no dualities with type preserving automorphism groups isomorphic to Sym(n) or Alt(n). We also develop general methods that extend to other settings. The residues of the two families are fundamentally different. In the first family, the maximal parabolics have constant size while in the other their size grows essentially linearly with the degree of the group.


Quasi-polynomials are ubiquitous in combinatorics and algebra, as they arise in a variety of enumeration problems. Because quasi-polynomials consist of constituent polynomials, their behavior is more subtle than for a single polynomial. In particular, unlike for a polynomial, it is possible for a quasi-polynomial defined on the positive integers to have infinitely many points at which it is decreasing. In this work, we characterize quasi-polynomials of degree $d$ and period dividing $p$ that are eventually nondecreasing, i.e., that have only finitely many values at which they decrease. We then give a detailed analysis of the space of eventually nondecreasing quasi-polynomials with fixed degree $d$ and period dividing a fixed $p$ such that the $h$-vector of the quasi-polynomial is nonnegative. Using this analysis, we determine the rate of growth of the number of such quasi-polynomials as a function of the sum of the $h$-vector entries for the $0$-th constituent polynomial.


Homogeneous hypergraph regularity lemmas via $k$-strong honest definitions

We prove that $(k+1)$-uniform hypergraphs definable in an NIP strongly $k$-distal structure satisfy a homogeneous regularity lemma -- they can be partitioned into a bounded number of simplicial complexes, most of which are homogeneous (meaning that the restriction of the hypergraph to the simplicial complex is either complete or empty). Furthermore, the parts of the partition can be chosen uniformly definably, and the size of the partition is polynomial in the reciprocal of the error parameter. This extends the homogeneous regularity lemma proven by Chernikov and Starchenko for hypergraphs definable in a distal structure.   We prove this by introducing $k$-strong honest definitions and showing that an NIP structure is strongly $k$-distal if and only if every formula $\varphi(x_1, ..., x_k; y)$ has a $k$-strong honest definition. This extends the theory of strong honest definitions in distal structures to the higher-arity setting.


On the autotopism groups and the equivalence of finite cyclic semifields

Special cases of finite cyclic semifields were first constructed by Hughes and Kleinfeld in 1960, and later by Sandler in 1962 and Knuth in 1965. The general construction of cyclic semifields was subsequently introduced by Petit in 1966, and later rediscovered from the perspective of irreducible semilinear transformations by Jha and Johnson in 1989. Since Sandler's foundational work in 1962, the complete determination of the autotopism groups of cyclic semifields and the full resolution of the isotopy problem for this family have remained long-standing open problems. The most significant advances in determining these autotopism groups are due to Dempwolff in 2011, who left open the case in which the field extension degree strictly divides the degree of the polynomial defining the semifield.   In this paper, we provide a complete classification of cyclic semifields up to isotopy, together with the full determination of their autotopism groups, thereby closing the remaining cases left open by Dempwolff. Since cyclic semifields arise as a special instance of a broader family of maximum rank distance (MRD) codes constructed via skew polynomials, our methods also yield a complete classification of these MRD codes up to linear and semilinear equivalence over the prime field, together with an explicit description of their full automorphism groups.


On the Slice Rank of Tensors in P-Echelon Form

For a totally ordered finite set $A$ with the total order $Q$, a poset $P=([d],\le_P)$, and a field $\mathbb{F}$, a tensor $T:A^d\longrightarrow \mathbb{F}$ is in $P$-echelon form if $r\le_{P}s$ in $P$ implies $a_r\le_Q a_s$ in every tuple $(a_1,\dots,a_d)$ in the support of $T$. We prove that if the Hasse diagram of $P$ has no isolated vertex, then tensors in $P$-echelon form with nonzero diagonal entries have full slice-rank. Our results extend and improve on recent results of Amanov and Yeliussizov.


Permutations with Verma Multiplicities $[M(p):L(q)] \geq 2$

We consider permutations $q$ in the symmetric group $S_n$ whose Verma multiplicities in the principal block of $\mathcal{O}(\mathfrak{sl}_n)$ satisfy $[M(p):L(q)] \geq 2$. We present a construction along with a diagrammatic visualization, showing how permutations in $S_n$ with this property generate a family of permutations in $S_{n+1}$ that share the same multiplicity property. While the method does not recover all such permutations in $S_{n+1}$, it systematically generates many new examples. In addition, we present all permutations in $S_5$ with $[M(\mathrm{id}):L(q)] \geq 2$ in a Bruhat diagram and describe all permutations in $S_6$ and $S_7$ with non-simple Verma multiplicities.


An $e$-positive classification for complete multipartite graphs

Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~$G=K_{(3,\,2^β)}$ are $e$-positive. We resolve this question and, together with their classification, characterize all $e$-positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$-expansion of~$X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$-coefficients of arbitrary complete multipartite graphs from the elementary--monomial Cauchy identity. For the particular graph~$G$, this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.


Various Bialgebras Of Representative Functions On Free Monoids

Factorization and decomposition of representative functions on a free monoid X * (generated by an alphabet X ) and with values in a ring A containing Q are equivalent to factorization and decomposition of their graphs (within the A-algebra of rational noncommutative series over X ) admitting linear representations (thanks to the Kleene-Sch{ü}tzenberger theorem).  To factorize and to decompose effectively these graphs, we examine various products of noncommutative series (as concatenation, shuffle and its $φ$deformations) and co-products such that, for A is a field K, their associated non graded commutative and co-noncommutative bialgebras of series are isomorphic to the Sweedler's dual of the graded noncommutative co-commutative bialgebras of polynomials having, for the concatenation, only Kleene stars of the planes as characters, or equivalently, only the planes are infinitesimal characters (thanks to a Ree's theorem like).


Counting spanning quasi-trees of ribbon graphs: determinants and #P-completeness

A quasi-tree of a connected ribbon graph is a spanning ribbon subgraph with exactly one boundary component; quasi-trees play the role of spanning trees in the topological graph theory of embedded graphs. We prove that counting them is #P-complete under polynomial-time Turing reductions, already for bouquets. The proof identifies every nonempty framed chord diagram, up to natural identifications, with a 4-regular map equipped with a distinguished A-trail, in such a way that quasi-trees correspond to A-trails, whose counting is #P-complete by a theorem of Ge and Štefankovič. Through the framed Cohn-Lempel equality the count is also an interlace-polynomial evaluation - $q(H;2,1)$, the number of full-rank induced subgraphs of the looped circle graph $H$ of the diagram - placing it on the line $y=1$ left open in the complexity classification of Bläser and Hoffmann; a cloning argument then makes every fixed rational point of that line, other than the trivial $(1,1)$, #P-hard on looped circle graphs, even when a framed chord representation is supplied. On the tractable side, the same GF(2) model yields short proofs of the known determinantal cases: for orientable ribbon graphs the count is a determinant, essentially the Matrix-Quasi-tree Theorem of Merino, Moffatt and Noble, proved here via Bouchet's principal unimodularity, and for bouquets with exactly one non-orientable loop it is a sum of two orientable determinants, equivalent by a rank-one determinant identity to the determinant formula of Deng, Jin and Yan.


Cographs and Minimum Diamond-Generating Edge Sets in Boolean Lattices

We study a local closure operation on the cover edges of a Boolean lattice: whenever the two lower edges or the two upper edges of a square face are present, all four edges of that square are added. We prove that every set of cover edges generating the full cover graph of $\mathcal{B}_n$ has cardinality at least $n$, and we classify all generators attaining this bound. For a graph $G$ on $[n]$, let $S_G=\{N_G(i)\to N_G(i)\cup\{i\}:i\in[n]\}$. Then $S_G$ diamond-generates the full cover graph if and only if $G$ is a cograph, and every minimum-cardinality generator arises uniquely in this way. Consequently, labeled minimum diamond-generating sets of $\mathcal{B}_n$ are in bijection with labeled cographs on $n$ vertices.


2-Connected Subgraphs of All Orders in Large Graphs with Minimum Degree at Least n/q

We confirm a conjecture of Liu and Ning\cite{LiuNing}: for every fixed integer $q\ge 3$, there exists an integer $n_0(q)$ such that every $2$-connected graph $G$ of order $n\ge n_0(q)$ with minimum degree $δ(G)\ge n/q$ contains a $2$-connected subgraph of order $\ell$ for each $\ell\in\{4,5,\ldots,n\}$. The proof has two main components. First, we construct a small $2$-connected subgraph $D$ such that every vertex outside $D$ has at least two neighbors in $D$. By successively adding the vertices outside $D$, we obtain $2$-connected subgraphs of every order from $|V(D)|$ to $n$. Second, an averaging argument on common neighborhoods yields a large complete bipartite subgraph $K_{2,t}$, which provides $2$-connected subgraphs of all the remaining orders.Finally, we propose the following conjecture: for every pair of fixed integers $r\ge 2$ and $q\ge 3$, every $r$-connected graph $G$ of sufficiently large order $n$ with $δ(G)\ge n/q$ contains an $r$-connected subgraph of every order from $2r$ to $n$.


Stretched exponential asymptotics for bases of triangular bootstrap percolation

In this paper, we study a bootstrap percolation process on the finite triangular grid $\mathfrak{T}_n$ of side length $n$. We say that a subset $η$ of points in $\mathfrak{T}_n$ percolates if the final configuration, starting from $η$, is the whole grid $\mathfrak{T}_n$. A basis of size $n$ is then a subset of points of $\mathfrak{T}_n$ of minimum cardinality which percolates. In this paper, we first prove that the generating function counting bases satisfies an algebraic differential equation. Then, by analysing a modified version of this equation, we prove that the number $t_n$ of bases of size $n$ exhibits a stretched exponential asymptotic behaviour. More precisely, we show that $t_n \sim c n!e^{\sqrt{12n}}n^{5/12}$, for some constant $c>0$. These bases were recently shown by the second author to be in bijection with $3$-permutations avoiding the patterns $(12, 12)$ and $(231, 312)$, so this represents to our knowledge the first proven example of an asymptotic stretched exponential appearing in the study of pattern avoiding permutations.


On Diophantine $m$-tuples related to primitive elements of finite fields

Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}$ be the multiplicative cyclic group of all non-zero elements over $\mathbb{F}_q$. An element $g\in\mathbb{F}_q$ is called primitive if $g$ generates the group $\mathbb{F}_q^*$. A set $\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*$ of $m$ elements is said to be a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ if $x_ix_j+1$ is primitive for any $1\le i\le j\le m$. Let $N_m$ denote the number of $\mathcal{P}$-Diophantine tuples over $\mathbb{F}_q$. Then we obtain the asymptotic formula   $$m!\cdot N_m=\left(\frac{\varphi(q-1)}{q-1}\right)^{m(m+1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right),$$   where $\varphi(\cdot)$ is the Euler totient function and $r\in(0, 1/2)$ is an arbitrary real number. Moreover, we prove that there exists a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ whenever $q\ge \exp(\exp(m(m+1)))$.


On the Computational Complexity of Local and Global Covering Numbers

The global and local $\mathcal{G}$-covering number $c_{\mathrm{g}}^{\mathcal{G}}(H)$ and $c_{\mathrm{l}}^{\mathcal{G}}(H)$ encode how well the edges of a graph $H$ can be covered with graphs from a graph class $\mathcal{G}$: in the global setting, we minimize the number of graphs from $\mathcal{G}$ required, in the local setting how often a vertex is hit by the graphs of the cover. Within this work we consider for $\mathcal{G}$ the graph classes $\mathcal{B}$ of all bipartite and $\mathcal{B}_{\mathrm{c}}$ of all complete bipartite graphs. We give a tight lower bound on $c_{\mathrm{l}}^{\mathcal{B}}(H)$ in terms of the fractional chromatic number of $H$, thereby giving a local analogue of a result by Harary, Hsu and Miller. Answering a question by Fishburn and Hammer, we show that it is NP-hard to determine $c_{\mathrm{l}}^{\mathcal{B}_{\mathrm{c}}}(H)$. Further, we provide a finite and monotone graph class $\mathcal{G}$ such that $c_{\mathrm{g}}^{\mathcal{G}}(H)$ can be computed in constant time for every graph $H$ while determining $c_{\mathrm{l}}^{\mathcal{G}}(H)$ is NP-hard. This yields a natural example to a question raised by Knauer and Ueckerdt.


Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma

Dittert's conjecture asserts that, among nonnegative $n\times n$ matrices whose entries sum to $n$, the functional $φ(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\operatorname{per}(A)$ is uniquely maximized by the uniform matrix $J_n/n$. This paper proves the conjecture for $n=16$. The key observation is that, for a near-maximizer, the deficits of the row-sum and column-sum products satisfy a single joint constraint rather than two independent bounds. Combining this joint-deficit estimate with a Pinsker-type subset-sum bound yields a sharper scalar dilation to a doubly superstochastic matrix. The Knopp-Sinkhorn boundary lower bound for permanents then excludes maximizers with a zero entry, and Hwang's positive-support theorem identifies the unique maximizer. Together with Pang's result for $n\ge 17$ (arXiv:2606.01531), this establishes Dittert's conjecture for every $n\ge 16$.


A classification of locally-quasiprimitive circulant digraphs

Circulant digraphs are Cayley digraphs over finite cyclic groups and constitute a fundamental class of objects in algebraic graph theory. Extending the classification of locally-primitive circulant graphs \cite{JZ-2026}, we completely determine all locally-quasiprimitive circulant digraphs. Our main theorem shows that a connected locally-quasiprimitive circulant digraph is isomorphic to one of the following: the complete graph \(\K_n\), the complete bipartite graph \(\K_{n/2,n/2}\), the graph \(\K_{n/2,n/2}-\frac{n}{2}\K_2\) (with \(n/2\) odd), the cycle \(\C_n\), the directed cycle \(\vec \C_n\), a normal circulant digraph of prime valency, the lexicographic product \(\vec \C_m[\overline{\K_b}]\), or the tensor product \(\vec \C_m\times \K_b\) with \(\gcd(m,b)=1\).


Polynomiality of Stretched Schubert Structure Constants and Key Coefficients

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. The proof combines Demazure operators with vector partition functions and, in the Schubert case, P.~Magyar's orthodontic formula. These coefficient results extend to finite products. Using M.~Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial, proving a conjecture of I.~Pak and Z.~Slonim. For key polynomials, this resolves the polynomiality part of a conjecture of P.~Alexandersson and E.~Alhajjar.


On the possible values of the Rearrangement Number

The rearrangement number $\mathfrak{rr}$ is the least cardinality of a collection of permutations of $ω$ such that every conditionally convergent real series is disrupted by some permutation in the collection. Blass, Brendle, Brian, Hamkins, Hardy, and Larson proved that $\max\{\operatorname{cov}(\mathcal N),\mathfrak b\}\leq\mathfrak{rr}\leq\operatorname{non}(\mathcal M)$ and asked whether $\mathfrak{rr}<\mathrm{non}(\mathcal M)$ is consistent.   We prove that $\mathfrak{rr}<\operatorname{non}(\mathcal M)$ is consistent with ZFC. We also prove, in a different forcing extension, that $\max\{\operatorname{cov}(\mathcal N),\mathfrak b\}<\mathfrak{rr}$. We further derive consequences for the subseries number $\mathfrak{s}_{\mathrm{sub}}$ and the splitting number $\mathfrak s$.


On the toric lifting properties for simplicial $3$-spheres

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.


A Local Classification of Four-Element Multiple Sumsets

For a finite set $A\subset\mathbb{Z}$, write $hA$ for its $h$-fold sumset, and let \[ R(h,k)=\{|hA|:A\subset\mathbb{Z},\ |A|=k\}. \] We determine the part of $R(h,4)$ lying between $4h+2$ and $6h-4$: for $h=4$ the only value is $5h-1$, while for $h\geq 5$ the only values are $5h-1$ and $5h+1$. This proves Rajagopal's conjectured gap $5h\notin R(h,4)$ for every $h\geq 4$. For $h\geq 6$, it also yields the new missing interval $[5h+2,6h-4]$, which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets of affine diameter five, of which there are only six. Reflection and four elementary exact sumset computations finish the classification.


A duality of misère games and play with a pass

In combinatorial game theory, the choice of play convention is a fundamental aspect of the theory. The two most widely studied conventions are normal play, in which the player who makes the last move wins, and misère play, in which the player who makes the last move loses. Another well-known variant is play with a single shared pass. In such games, at most one pass may be used in total during the game: once the pass has been used, neither player may pass thereafter. Moreover, once a terminal position has been reached, passing is no longer allowed. Recall that, for misère play, one sometimes defines SG values (or Sprague-Grundy values) in the same recursive manner as in normal play but assigns the terminal position the value 1. Also, SG values can be considered for normal-play games with a pass. In this work, we propose transformations that allow both misère play and play with a pass to be treated as normal-play games. As a consequence, we show that there is a certain duality between a generalization of misère play and a generalization of play with a pass.


Regularity and depth of binomial ideals arising from combinatorics

Regularity and depth of binomial ideals generated by adjacent $2$-minors together with those arising from finite lattices are studied.