Papers

Several important fans studied at the interface of combinatorics and algebraic geometry arise from barycentric subdivisions of other fans; the central example is the braid arrangement. The collection of faces of the standard simplex forms a basis for the deformation cone of the braid arrangement, i.e. the cone of generalized permutahedra. We reinterpret this simplicial basis as the collection of deep truncations of the standard simplex, and systematically abstract this perspective to produce polytopal bases for the deformation cones of barycentric subdivisions of simplicial projective fans. We further demonstrate that these bases restrict to bases for deformation cones of fans obtained by a sequence of stellar subdivisions induced by a building set. When the starting polytope is smooth with all edge lengths equal to one, we show that we can upgrade our basis to a collection of flat truncations. We then investigate two extensions of the braid arrangement where we are able to further upgrade our flat truncation basis to an indecomposable polytopal basis: 1. the barycentric subdivision of the normal fan of a product of standard simplices and 2. the barycentric subdivision of the braid arrangement. We discuss connections to, and implications for, Archimedean solids and regular polytopes, permutahedral plates, root polytopes, permutoassociahedra, simple permutonestohedra, cosmohedra, omnitruncations of Coxeter permutahedra, bipermutahedra, $\pi$-colored fans, and tropical $\alpha$ and $\beta$ classes.

Let $\mathcal{B}$ be the class consisting of the six-vertex bipartite graphs that possess a perfect matching and their complements. It is proved that every $\mathcal{B}$-free graph $G$ satisfies $\alpha(G)+\omega(G)\ge |V(G)|-1$. This establishes Conjecture 3.1 of Litjens, Polak and Sivaraman (B-Free Graphs Conjecture). For a smallest counterexample, Hall-type exchange arguments show that two maximum stable sets, and likewise two maximum cliques, differ in at most two vertices. A core-corona matching lemma then forces $|\alpha(G)-\omega(G)|\le 2$. Double counting between suitably dense and sparse vertices reduces the problem to twenty-one binary feasibility systems on at most fourteen vertices. Their infeasibility is verified by two independent exact encodings, with a separate exhaustive validation of the forbidden-family constraints.

Suppose $X,Y$ are independent random variables with values in a compact abelian group $(G,+)$. We examine the following two entropy power-type inequalities: $h(X+Y)\geq \frac{1}{2}h(X)+\frac{1}{2}h(Y)$ and $h(X+Y)\geq \max\{h(X),h(Y)\}$, where the entropy $h(Z)$ of a $G$-valued random variable $Z$ is defined in terms of its density with respect to Haar measure on $G$. For groups that are either connected or finite with no nontrivial subgroups, we precisely characterize the cases of equality and establish explicit, quantitative stability estimates in terms of relative entropy for these two inequalities. The main tools are a generalization of an entropic inequality obtained by Green, Manners and Tao (2023) for discrete entropy, and a harmonic-analytic estimate for the chi-squared contraction coefficient in connected compact groups. As an application, we derive exponential convergence rates to the uniform distribution in relative entropy for random walks on connected compact abelian groups.

The Hadamard maximal determinant problem seeks an $n\times n$ matrix $X$ with entries in $\{\pm 1\}$, which maximizes the determinant $\vert \det X \vert$ for a given order $n \in \mathbb{N}$. We report matrices attaining new record determinants for orders $n\equiv 3\pmod{4}$ between $103$ and $119$, as well as $n=51$. We provide a proof of optimality within a specific family of circulant–block constructions for $X$. For these orders, we recast the Hadamard problem as a sequence reconstruction problem from a pair of periodic autocorrelation functions, subject to certain arithmetic conditions. We describe several learning–based strategies for reconstruction and provide a comparison with classical simulated annealing–style approaches as a benchmark.

Baste et al. (2020) conjectured that every regular graph of positive degree has domination number at most its edge domination number, the least size of a maximal matching. Combining published bounds settles the inequality for every degree at least nine. A reduction proves the inequality whenever one endpoint of each edge of a minimum maximal matching can be chosen to form a dominating set, and the Lovász Local Lemma shows such a choice exists for every degree at least seven, newly closing degrees seven and eight and leaving degrees three through six open. The reduction settles each open degree up to a bounded number of vertices, forty-eight for cubic graphs. At fifty vertices, however, the reduction meets an explicit cubic graph it cannot settle, though the inequality holds there too. The inequality cannot be tightened, since infinitely many cubic graphs have equal domination and edge domination numbers. The cubic case stays open, and even linear arguments from the local structure cannot close it. The middle degrees stay open beyond the graphs already settled.

The queen's graph $Q_n$ has the squares of the $n\times n$ chessboard as vertices, with adjacency defined by a common row, column, or diagonal. We give an explicit set of $82$ queens on $Q_{163}$. In centered coordinates, all queen coordinates are odd, every odd row and odd column is occupied exactly once, and the occupied rows, columns, and diagonals satisfy the conditions for a type A $1$-cover in the terminology of Ostergard and Weakley. A direct independent verification checks every one of the $163^2=26{,}569$ board squares and finds none uncovered. Hence $\gamma(Q_{163})\leq82$. The Finozhenok-Weakley lower bound $\gamma(Q_n)\geq\lceil n/2\rceil$, valid here, gives the matching inequality and therefore $\gamma(Q_{163})=82$. For this cover, the parameters defined by Ostergard and Weakley are $e=16$, $f=15$, and $u=24$; the complete difference- and sum-diagonal multisets are displayed in the paper. It consequently also supplies an explicit finite input to their amplification theorem for type A covers, giving $\gamma(Q_N)\leq(17/33)N+O(1)$. This last coefficient improves both the earlier type A coefficient $69/133$ and the subsequent general coefficient $101/195$ of Burger and Mynhardt. The coordinates and a complete standard-library Python verifier are included.

Let $N=2^k-1$, and let $S_{t,k}$ consist of the pairs $0\le a,b<N$ such that $a+b\equiv t\pmod N$ and $\wt(a)+\wt(b)<k$. The Tu–Deng bound $|S_{t,k}|\le 2^{k-1}$ has recently been proved. In this paper, we give a combinatorial proof of the equality criterion. If the $k$-bit cyclic word of $t$ has $Z$ zeros and $g_1,\ldots,g_Z$ are the numbers of ones between successive zeros, then $$|S_{t,k}|=2^{k-1} \quad\Longleftrightarrow\quad g_i\ge Z-1\quad(1\le i\le Z).$$ Moreover, we recover the resulting enumeration of the equality parameters. Beyond equality, if $R\ge Z\ge2$, where $R$ is the number of ones, then every nonequality parameter satisfies $$2^{k-1}-|S_{t,k}|\ge 2^{R-Z+1},$$ and we classify all cases in which this bound is attained. For $R<Z$ we obtain a congruence for $|S_{t,k}|$ and a lower bound for the deficit in terms of the number of cyclic runs of ones.

In this article, we prove two conjectures of Burness and Fusari [Timothy Burness and Marco Fusari, On derangements in simple permutation groups, Forum Math. Sigma 13 (2025)] concerning the powers and products of conjugacy classes of derangements in the symmetric and alternating groups: (1) We show that there exist two conjugacy classes $C$ and $D$ of derangements in $S_n$ such that $S_n=C^2\cup CD$, and (2) We show that there exists a conjugacy class $C$ of derangements in $A_n$ such that $C^2=A_n$, whenever $n\equiv 3\;(\text{mod}\;4)$. In fact, our result concerning the second conjecture holds in a considerably more general setting, which also answers affirmatively a question posed by Bertram [Edward Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory, Ser. A 12 (1972), 368-380] in a particular case. Moreover, we show that any conjugacy class $C$ of derangements in $S_n$ (resp. $A_n$) contains a pair of elements that generate $S_n$ or $A_n$ (resp. $A_n$), unless $C$ is the conjugacy class of fixed-point-free involutions.

Given an integer $k\geq2$ and a graph $F$, the $k$-uniform suspension $\mathcal{S}^kF$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K^k_t$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ and $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\mathcal{S}^kC_{2\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\geq2$ and infinitely many $n$, \[\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).\] This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{ex}(n,\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\geq2$ and $\ell\in\{3,5\}$, we determine the order of magnitude \[\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=\Theta(n^{k-1+1/\ell}).\] This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\ell=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ with $\ell\in\{3,5\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.

A threshold graph is a graph that can be constructed from the one-vertex graph by repeatedly adding either a dominating vertex or an isolated vertex. Motivated by an induced Ramsey-type problem for this class, we define $r'_2(s)$ to be the minimum integer $n$ such that every $n$-vertex graph contains an induced threshold graph on $s$ vertices. We establish exponential upper and lower bounds for $r'_2(s)$ and determine its exact values for $s\in\{3,4,5,6\}$. To study this problem from an edge-coloring perspective, we use the notion of an orderable coloring, introduced by Richer [J. Combin. Theory Ser. B, 80(1) (2000), 172–177]. An edge-colored graph is orderable if its vertices can be ordered so that, for each vertex, all edges from it to later vertices have the same color. Equivalently, $r'_2(s)$ is the minimum $n$ such that every $2$-edge-coloring of $K_n$ contains an orderable $K_s$. We also determine the exact value of the unordered canonical Ramsey number $CR(s, 3)$ for all $s \ge 3$, where $CR(s,3)$ denotes the minimum integer $n$ such that every edge-coloring of $K_n$ contains either an orderable $K_s$ or a rainbow $K_3$. More generally, for graphs $G$ and $H$, we study $r'_2(G)$, the corresponding $2$-color Ramsey number for an orderable $G$, and $CR(G,H)$, where the alternative is a rainbow $H$. For complete bipartite graphs, we prove that for every fixed $s$, $r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$ as $t\to\infty$. For $s\in \{2,3\}$, we further determine the exact values of these parameters for infinitely many $t$, using constructions arising from strongly regular graphs, Hadamard matrices and conference matrices.

The Erdős-Pósa theorem states that for every integer $k\geq1$, every graph contains either $k$ vertex-disjoint cycles or a set of $\mathcal{O}(k\log k)$ vertices meeting all cycles. This fundamental min-max duality has been extended to numerous settings, including long cycles, $S$-cycles, that is, cycles containing a vertex in a prescribed set $S$, and cycles satisfying various additional constraints. In contrast, much less is known when the packing itself is required to be induced, namely, when distinct cycles are vertex-disjoint and have no edges between them. We prove that long $S$-cycles admit an induced version of the Erdős-Pósa-type duality. More precisely, we show that there exists a polynomial function $f(k,\ell)$ such that for all integers $k\geq1$ and $\ell\geq3$, every graph contains either an induced packing of $k$ $S$-cycles of length at least $\ell$ or a set of at most $f(k,\ell)$ vertices whose closed neighbourhood intersects all $S$-cycles of length at least $\ell$. The proof introduces a new ear-decomposition technique based on fragile ears and yields a polynomial-time algorithm for every fixed $\ell$.

For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$-point sets that improve the best known upper bounds for $H_{a,d}(n)$ in several cases of $a$ and $d$. We also study a dual version of the problem. Let $D_d(n)$ the maximum number such that for any arrangement of $n$ hyperplanes in general position in $\mathbb{R}^d$, we can always find a subset of $D_d(n)$ hyperplanes for which all the $d$-dimensional simplices that they define have distinct $d$-dimensional volumes. We improve the current known upper bound for $D_d(n)$ and give the first nontrivial lower bound for $D_2(n)$ and $D_3 (n)$.

The Minimum Matching Problem consists of finding an independent edge set of minimum weight $M_{\star}(G)$ in a given edge-weighted graph $G$. When $G$ is bipartite, this reduces to the Assignment Problem. We consider a variant of this problem defined by taking the union of optimal matchings across various slightly modified versions of the base graph: $H_{\mathcal{J}}(G)=\bigcup_{U \in \mathcal{J}} M_{\star}(G_{U})$. We establish two families of results: (1) In two distinct settings for the Assignment Problem, we prove that the resulting graphs $H_{\mathcal{J}}$, as well as certain associated graphs $\bar{H}_{\mathcal{J}}$, are spanning trees on the relevant base graphs $G$ and $\bar{G}$. (2) In these same settings, assuming the edge weights are given by the $p$-th power of Euclidean distances for point configurations in the plane, we show that for $p=1$ the tree $H_{\mathcal{J}}$ is non-crossing (i.e., its planar embedding has no crossing edges), whereas, remarkably, for $p=2$ the associated tree $\bar{H}_{\mathcal{J}}$ is non-crossing. Finally, we introduce novel conjectures in Statistical Mechanics, to be explored in future work: in the Random Euclidean Assignment Problem (where points are i.i.d.\ on a planar domain), we conjecture that for $p=2$ the trees $\bar{H}_{\mathcal{J}}$ are asymptotically distributed as Uniform Spanning Trees with free and wired boundary conditions in the two respective settings. In particular, suitable paths on the tree in the second setting, and on its planar dual in the first setting, are asymptotically distributed as $\text{SLE}_{\kappa}$ with $\kappa=2$.

The aim of this paper is to introduce pairing theory for partitions. We begin with two statistics on integer partitions, the pairing index and the pairing rank. The pairing index is equidistributed with the number of parts, while a joint refinement identifies its two constituents with the numbers of even and odd parts. We further introduce the pairing width and prove that pairing index and pairing width are jointly equidistributed with the number of parts and the largest part. The resulting finite Gaussian generating function has a cyclotomic factorization from which Kummer's famous carry theorem for binomial coefficients follows. We also prove a mod-$5$ congruence for the excess of unpaired parts congruent to $1$ modulo $4$ over those congruent to $3$ modulo $4$ in the partitions of $5n+4$. A signed specialization exhibits that the parity of the pairing rank is governed by self-conjugate partitions. Motivated by this, we go on to introduce a second, diagrammatic pairing: after the two wings of the Durfee square are folded together, the unpaired cells break into connected diagonal blocks. These blocks may be reflected independently, giving a Boolean decomposition of the set of partitions with a unique representative having all successive ranks nonnegative. We then relate our theory to overpartitions and Frobenius representations, obtaining as a corollary a geometric realization of overpartitions in terms of partitions whose principal hooks are all even. Finally, we study simply paired partitions of negative pairing rank, obtaining identities involving odd divisors and overpartitions, a parity theorem for pairing rank $-2$, and a Toeplitz determinant whose coefficientwise limit is an explicit infinite product related to MacMahon's product for plane partitions.

We show that approximately counting the vertices of a bounded 0/1 polytope, presented as a system of rational linear inequalities, is, informally speaking, NP-hard. In particular, there is no FPRAS for this problem unless RP=NP. The proof is by a reduction from approximately counting homomorphisms from a given graph to a particular four-vertex graph. The main proof ideas were found using GPT-5.6 Sol Ultra.

We use the Pfaffian minor summation formula together with Jacobi–Trudi Toeplitz matrices to derive Pfaffian expansions in skew Schur functions. This yields explicit Schur expansions for several generating functions involving products of skew Schur functions. In particular, using sparse skew-symmetric matrices, we provide a Pfaffian proof of a Schur-positive identity arising in the study of the $q$-log-convexity of the Narayana polynomials. As the main application, we prove that the Baxter polynomials form a $q$-log-convex sequence. We further show that the Baxter transformation defined by the refined Baxter numbers preserves log-convexity. Finally, by realizing the $q$-refined Baxter numbers as principal specializations of rectangular Schur functions, we prove that the array of $q$-refined Baxter numbers is $q$-log-concave both along each row and along each column. Both $q$-log-concavity results extend naturally to the $q$-analogues of the $d$-Hoggatt numbers.

Let ${T}_n$ be a random $2$-dimensional determinantal hypertree on $n$ vertices. Given any prime $p$, we answer the following question: If a cocycle in $Z^1({T}_n,\mathbb{F}_p)$ has small support, what does the support typically look like? More precisely, we characterize all the finite connected graphs $G$ for which there is a constant $c_G>0$ with the following property: For all large enough $n$, with probability at least $c_G$, we have a cocycle $f\in Z^1({T}_n,\mathbb{F}_p)$ such that after removing all the isolated vertices, the support of $f$ is isomorphic to $G$. We prove that for $p>2$, we do not have any such graph. For $p=2$, a connected graph has the property above if and only if it has a unique cycle such that this unique cycle has odd length, moreover, if the unique cycle is a triangle, then we also need to require that all the vertices of the triangle have degree at least $3$.

We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over $\mathbb{F}_2$, which gives us the number of algebraic tori necessary to cover the cluster manifold over $\mathbb{F}_2$. We also show that these formulae satisfy certain recurrence relations.

Determining when the chromatic symmetric function of a graph is Schur positive is a central problem in algebraic combinatorics. Elementary expansions can help answer this question, but negative coefficients obstruct a direct positivity argument. We develop methods for proving Schur positivity from signed elementary expansions and apply them to chromatic symmetric functions of clique-spiders and three-legged spiders. We first give the Schur alpha-omega lemma, which bounds Schur support. We then give four equivalent sufficient criteria for Schur positivity from a fixed composition-indexed elementary expansion. The matrix, order-ideal, negative Hall, and positive Hall formulations treat positive coefficients as capacities and negative coefficients as demands, with admissible assignments governed by dominance order. These criteria provide certificates of Schur positivity especially when elementary coefficients are negative. For the four basic families of $3$-clique-spiders with total leg length at most one, we obtain complete classifications of both $e$-positivity and Schur positivity. These classifications show how the position of a single leg edge changes positivity. Using selected clique-spider cases and a path-clique bootstrap, we prove that all spiders of the form $S(a,b,2)$ are Schur positive.

Given a word over a finite alphabet, consider the sequence that counts, for each length, the number of distinct subsequences of that length. In 1976, Chase proved that this sequence is log-concave. His proof uses a triangular array indexed by the prefixes of the word together with a meticulous analysis of ratios of several sums. We instead decompose according to the first letter, which reduces the proof to a weighted average.

Let $f(n)$ be the least possible largest element of an $n$-element set of positive integers with pairwise distinct subset sums. Dubroff, Fox and Xu proved the finite lower bound \[ f(n)\ge \binom{n}{\lfloor n/2\rfloor} \] by applying a vertex-boundary estimate to the $2^{n-1}$ subsets whose sums lie below half of the total sum. We instead order all $2^n$ subsets by increasing subset-sum value. We prove that the bandwidth of this numbering is at most the largest element of the set, so the exact formula for the bandwidth of the hypercube gives the stronger finite bound \[ f(n)\ge H_n:=\sum_{j=0}^{n-1}\binom{j}{\lfloor j/2\rfloor}. \] An exact identity for $H_n$ in terms of Catalan numbers yields \[ \frac{H_n}{\binom{n}{\lfloor n/2\rfloor}} =\begin{cases} 1+\dfrac{2}{3n}+O(n^{-2}),& n\text{ even},\\[5pt] 1+\dfrac{4}{3n}+O(n^{-2}),& n\text{ odd}. \end{cases} \] For every $n\ge3$, this bound is strictly larger than the central-binomial bound. As a separate corollary, a first-order asymptotic formula for factorials rewrites this lower bound as a multiple of $\sqrt{2/\pi}\,2^n/\sqrt n$; the coefficients of $1/n$ are $5/12$ in even dimension and $7/12$ in odd dimension.

We prove that the energy ${\mathcal E}(G)$ of any simple graph $G$ of order $n\ge5$ satisfies \[ {\mathcal E}\ge r(G)+\bar d(G)-1, \] where $r(G)$ and $\bar d(G)$ denote, respectively, the rank of the adjacency matrix and the average degree of $G$. We also characterize all extremal graphs. As consequences, our result settles five previously conjectured lower bounds for the energy of nonsingular graphs in their stated ranges, namely \[ \begin{gathered} {\mathcal E}(G)\ge n-1+\bar d(G),\qquad {\mathcal E}(G)\ge\Delta(G)+\delta(G),\qquad {\mathcal E}(G)\ge2\sqrt{\bar d(G) (n-1)},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{m},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{2m}+\frac{2m}{n}, \end{gathered} \] where $m$ is the number of edges, $\Delta(G)$ and $\delta(G)$ are the maximum and minimum degrees, and the first Zagreb index $ M_1(G)$ is the sum of degree squares.

We show that, in the ring of virtual species, \[ \mathcal{I}=\sum_{m\geq 0}(-1)^m\prod_{i=1}^{m}\bigl((E^{-1})^i-1\bigr), \] where $\mathcal{I}$ is the species of interval orders and $E^{-1}$ is the multiplicative inverse of the species $E$ of sets. The right-hand side is the virtual species of signed ballot matrices introduced by Claesson and Hannah. They showed that its signed cardinality counts labeled interval orders. We strengthen this to a species identity, which we prove twice: first algebraically and then bijectively, using a natural sign-reversing involution. The cycle index series of $\mathcal{I}$ specializes to the generating series for labeled and unlabeled interval orders. We describe the automorphism group of an interval order as a Young subgroup and prove the identity $\mathcal{I}=\mathcal{R}\circ E_+$, where $\mathcal{R}$ is the species of rigid interval orders. We also show that Glaisher's T-number $T_n$ counts the $24$-colored interval orders on $[n]$ in which no isolated element has color $24$.

Let $F_k(G)$ denote the $k$-token graph of a finite graph $G$. The token graph reconstruction conjecture asks whether, for any $1\le k<|V(G)|$, $F_k(G)\cong F_k(H)$ implies $G\cong H$. This paper proves that the conjecture holds for almost every graph.

We prove seven conjectures involving binomial coefficients. These conjectures appear in The On-Line Encyclopedia of Integer Sequences (OEIS). Among these is an application of Jacobi's residue formula. We also apply MacMahon's master theorem to give a new and independently obtained proof of a recently established result.

A generalized Steinhaus $s$-triangle is obtained from a binary sequence by repeatedly replacing each block of $s$ consecutive entries by its sum modulo $2$. We study the triangles generated by canonical basis vectors. Expressing their entries through bi$^{s}$nomial coefficients, we prove that the truncated coefficient profiles of the successive rows are purely periodic and determine their exact period. This periodicity yields a power-of-two derivative identity and a decomposition into identical finite blocks. We consequently obtain an exact recurrence for the number of ones and show that, on every residue class, this weight is an affine function of the length. We also derive a rational generating function, determine the asymptotic growth rate, and prove that these canonical triangles have zero density of ones. Explicit formulas are obtained for the first two canonical vectors and for the block weight in the classical case. These results provide a unified description of the periodic and enumerative structure of generalized Steinhaus triangles generated by canonical basis vectors.

Many imaging problems seek to reconstruct underlying configurations from partial observable measurements. While reconstruction algorithms operate on these measurements, the observable organization induced by the measurement process is rarely represented explicitly. We introduce TENSKEL, a combinatorial observable framework for structured measurement and reconstruction based on tensor representations defined over discrete domains. Starting from a binary latent ensemble, the framework constructs a hierarchy of tensors coupling measurement contexts to a latent Pascal organization through successive aggregation and folding operations. Each measurement context induces an observable partition of the same latent ensemble, and the resulting tensor formulation makes explicit the associated combinatorial multiplicities, shell organization, degeneracies, and induced reconstruction geometry. Rather than introducing a new reconstruction algorithm, this framework provides a mathematical representation of how latent configurations become organized under observation. The induced tensor kernel characterizes similarities between latent coordinates through their measurement-context responses, while regularized inversion provides a structured reconstruction of the latent representation from observable measurements. The binary construction further admits a natural multinomial extension to discrete simplex-supported latent representations. Connections to Pascal cellular automata and structured discrete color mappings illustrate respectively compressed and multinomial realizations of the framework. More generally, TENSKEL provides a combinatorial basis for reasoning about the organization induced by observation, with potential relevance to computational imaging, inverse problems, structured sensing, and quantum-inspired measurement formulations.

We characterize infinitesimal finite forcibility for bounded symmetric real kernels. We prove that the graph-density gradients at a kernel span a finite-dimensional space if and only if the kernel is a step kernel. Combined with known finite-forcing results for step kernels, this gives a positive answer to a question of Lovász and Szegedy on whether every infinitesimally finitely forcible kernel is finitely forcible. The proof combines spectral methods with a compression argument based on book graphs.

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\ldots,v_n\}$, and let $\omega:V(G)\to \mathbb R^+$ be a positive vertex-weight function satisfying $\omega(v_i)=x_i$ for each $v_i \in V(G)$. The weighted Kirchhoff index of $G$ is defined by $K(G;x_1,x_2,\ldots,x_n)=\sum_{1\le i<j\le n}x_i x_j r_G(v_i,v_j)$, where $r_G(v_i,v_j)$ denotes the resistance distance between $v_i$ and $v_j$. In this paper, we give a combinatorial interpretation of the weighted Kirchhoff index of an arbitrary connected graph. More precisely, we express $K(G;x_1,x_2,\ldots,x_n)$ in terms of the sums of weights of matchings in an appropriately weighted subdivision graph of $G$, and in the subgraphs obtained from this weighted subdivision graph by deleting the subdivision graphs corresponding to \(2\)-regular subgraphs of $G$. This gives an affirmative answer to a question posed by Li, Li and Yan [Discrete Math. 345 (2022) 113109] concerning a combinatorial explanation of the weighted Kirchhoff index of a general graph by using matchings in weighted subdivision graphs and their subgraphs. As special cases, our formula recovers the known formulas for the weighted Kirchhoff index of trees and unicyclic graphs, as well as the known formula for the ordinary Kirchhoff index of an arbitrary connected graph.

Fox's conjecture predicts that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. Kálmán, Mészáros, and Postnikov gave a new proof of trapezoidality for special alternating links using a polynomial associated with flat vector arrangements. We prove that the flat arrangement polynomial has trapezoidal coefficients for every full-dimensional flat vector arrangement, answering the corresponding open question. Our proof gives a geometric interpretation of this polynomial in terms of convex polytopes. As an application, we prove that the Murasugi–Stoimenow polynomial of every connected Eulerian digraph has trapezoidal coefficients.