Papers

We develop an abstract axiomatic theory of tie-breaking. A tie-breaking input consists of a finite set N of players, a weak order on N representing the standings to be refined, and an auxiliary information item drawn from a set on which the symmetric group Sym(N) acts. Within this minimal framework we prove three theorems. First, no tie-breaking rule producing a strict linear order can be anonymous, provided the input space contains even one intrinsically symmetric situation, a condition met in essentially every realistic application. Second, when we allow the rule to output a partition of N (rather than a strict ranking), there is a unique rule satisfying two natural axioms: it is the partition of N into orbits of the joint stabilizer of the input. Third, every reasonable strict tie-breaking rule decomposes uniquely as the canonical orbit partition followed by an arbitrary completion. The decomposition makes precise the informal observation that real tie-breaking systems are honest until forced to be arbitrary. The framework is broad enough to capture chess tournament tie-breakers, sports league regulations, voting tie-breakers, tie-breaking among symmetric players in cooperative games, and ranking by network centrality measures, all within a single uniform formalism.

In this article, we investigate the statistical distribution and asymptotic behavior of the family of monic integer polynomials of degree $n$ having at least one root in a fixed number field $K$. Although the framework of thin sets implies that the natural density of this family in the parameter space of bounded height is zero, explicitly quantifying this vanishing rate is a central challenge in arithmetic statistics. Employing a hybrid approach that integrates the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function, we demonstrate that the rate of convergence of this density to zero is strictly dependent on the degree $n$. Specifically, we prove that the degrees of the factors induce a phase transition in the asymptotic behavior; for polynomials of degree $n = 2$, the decay rate is bounded by $O(H^{-1} \log H)$, whereas for higher degrees, the asymptotic behavior is dominated by the contribution of rational roots, yielding a bound of $O(H^{-1})$. Beyond deriving these asymptotic estimates, we apply principles from the geometry of numbers to establish explicit combinatorial bounds for counting both the reducible and irreducible components of these polynomials. These explicit bounds provide practical tools for computational evaluations within this domain.

In this short paper, we begin to investigate the conditions under which a generic Bipermutive Cellular Automaton (BCA) with no-boundary conditions of diameter $d$ generates a Latin square of order $N=2^{d-1}$ admitting an orthogonal mate, without relying on the linearity of the local rule. Since an orthogonal mate exists if and only if the Latin square can be partitioned into $N$ disjoint transversals, we start by characterizing the subclass of BCA whose Latin squares have a transversal on the main diagonal. In particular, we prove that the main diagonal forms a transversal if and only if the generating function of the bipermutive local rule induces an invertible CA with periodic boundary conditions on a configuration of size $d-1$. We then perform exhaustive search experiments, showing that $d=6$ is the smallest diameter for which there exist nonlinear bipermutive CA that generate Latin squares with a transversal on the main diagonal.

We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$, the $n$th moment can be interpreted as the number of abelian squares of length $2n$ over an alphabet with $m$ letters, for $d=4$ no interpretation was known. The goal of this paper is to provide such an interpretation, both for $d=2$ and $d=4$, in terms of $2n$-step lattice paths in dimension $m-1$. Our construction relies on a bijection between Dyck paths with a prescribed number of peaks and words of a certain type. In addition, this bijection allows us to derive closed formulas for the number of lattice paths provided with certain statistics.

We study crossing numbers of dense graph drawings whose vertices are uniformly distributed either on the unit sphere or in a compact convex planar domain. We prove a sharp inequality for weighted geodesic drawings on $\mathbb S^2$ in a continuous setting: among all measurable edge arrangements of a fixed density, the amount of crossings is minimized by connecting pairs of points within a fixed distance threshold. We also prove a planar analogue for straight-line drawings in convex planar domains. We transfer these continuous results to finite graphs using a smoothing argument. In the small density limit, we recover the conjectured midrange crossing constant lower bound of $8/(9\pi^2)$ for this restricted model.

We report on the status of the conjecture of Bousquet-Mélou and Mishna that the univariate counting generating function of a small-step quarter-plane lattice model is D-finite if and only if the group of the walk is finite. While the finite-group case is fully resolved, the infinite-group case remains incomplete. We list the arguments for the non-D-finiteness for 21 of the 56 infinite-group models: the five singular models, three models with zero drift and thirteen models with polar interior drift. The proof of the latter two families uses asymptotic results of Bostan–Raschel–Salvy combined with probabilistic estimates of Denisov–Wachtel and Duraj. We further identify nine infinite-group models whose endpoint counting series are differentially algebraic via decoupling functions, though this does not settle their D-finiteness. For 21 of the remaining models, numerical estimation of singular exponents suggests non-D-finiteness of one of its the boundary series $Q(1,0;t)$ or $Q(0,1;t)$, and we state a conjecture to this effect.

The packing chromatic number of a graph is the minimum number of colors for which the graph admits a packing coloring. This distance-based parameter may change under local structural modifications of the graph. In this paper, we introduce the packing coloring gap, defined as the maximum decrease in the packing chromatic number caused by the deletion of a single vertex. We focus on trees and determine the packing coloring gap for caterpillars. We further extend these results to caterpillars under the corona operation with K1. In addition, we present examples of graphs with packing coloring gap zero, one, and arbitrarily large.

We define a group of lower-triangular matrices whose columns are defined by power series. This group can be seen as a generalization of the (ordinary) Riordan group and the double Riordan group. Elements of this group are defined by three power series. Sequence bisections and vertically stretched Riordan arrays play an important role in the formulation of this group. We give a production matrix characterization of this new group. We also indicate how higher order groups can be defined, based on $n$-tuples of power series. We have chosen to name this group in memory of Renzo Sprugnoli, who was a pioneer in the application of the Riordan group to combinatorial problems as well as contributing to an understanding of the rich structure of Riordan arrays.

The classical recursive upper bound on hypergraph Ramsey numbers due to Erdős and Rado states that for $2 \leq k < s \leq t$, \[ r_k(s,t) \leq 2^{\binom{r_{k-1}(s-1,t-1)}{k-1}}. \] In 2010, Conlon, Fox, and Sudakov introduced the so-called vertex online Ramsey numbers $\tilde{r}(s,t)$ for graphs to obtain a quantitative improvement over this bound when $k=3$. In this note, we show that the natural hypergraph generalization $\tilde{r}_k(s,t)$ of the vertex online Ramsey numbers satisfy an improved recurrence \[ \tilde{r}_k(s,t) \leq 2^{(1+o(1))\tilde{r}_{k-1}(s-1,t-1)}. \] We obtain several corollaries from this, including a lower-order improvement to the best known quantitative upper bounds for hypergraph Ramsey numbers and an improvement to the above recursive bound of Erdős and Rado.

For the OEIS sequence A348410, P. Bala recorded in February 2022 two equivalent closed forms, $a(n) = [x^{n}] ((1-x)(1-x^2))^{-n}$ and a single-index binomial sum. R. J. Mathar (October 2021) and V. Kotesovec (November 2021) each contributed a conjectured P-recursive recurrence – Mathar's of order $4$, Kotesovec's of order $2$. We apply Lagrange-Bürmann inversion to Bala's $[x^n]$ form to derive the parametric expression $A(t) = (1 - y^2)/(1 - y - 4 y^2)$, where $y = y(t)$ is implicit by $y(1-y)^2(1+y) = t$. Eliminating $y$ via resultant gives the explicit algebraic equation $P(t, A) = 0$ of degree $4$ in $A$ and degree $2$ in $t$. As an immediate corollary (Stanley's classical algebraic-implies-D-finite theorem), $A(t)$ is D-finite. Mathar's and Kotesovec's specific recurrences are not directly proven here; we only verify Kotesovec's order-$2$ recurrence numerically for $n = 3, \ldots, 1000$ and observe that an explicit ODE-and-recurrence extraction from $P(t, A) = 0$ via the standard Bostan-Chyzak-Salvy algebraic-to-holonomic procedure would close both conjectures. The supplementary archive contains a SymPy script which derives $P(t, A)$ and checks the numerical evidence.

For a graph $G$ and integers $a_i \geq 1$, we say that $G \xrightarrow[]{} (a_1, \ldots, a_k)^v$ if in any $k$-coloring of $G$'s vertices there exists a monochromatic $a_i$-clique for some color $i \in \{1,\ldots,k\}$. $G \xrightarrow[]{} (a_1, \ldots, a_k)^e$ is defined similarly, but for edge colorings. The Folkman number $F_v(a_1, \ldots, a_k; H)$ is the smallest number of vertices for which an $H$-free graph arrowing $(a_1, \ldots, a_k)^v$ exists. $F_e(a_1, \ldots, a_k; H)$ is defined similarly for edge-arrowing. In this work, we present new bounds for Folkman numbers where $a_i \in \{2,3\}$ and $k \leq 4$, while avoiding $K_n$, $J_n$, for $n \in \{4,5,6\}$, where $K_n$ is the complete graph on $n$ vertices and $J_n$ is $K_n$ missing an edge. We also present results for $C_4$-free and $W_5$-free graphs, where $C_4$ is the cycle on four vertices and $W_5$ is the wheel graph on five vertices. Notably, we prove the existence of $F_e(3,3;W_5)$, leaving only one graph, $\overline{P_2 \cup P_3}$, on five vertices for which the existence problem of $F_e(3,3;H)$ remains open. We provide some theoretical results that should aid in uncovering the existence of $F_v(3,3; \overline{P_2 \cup P_3})$. Our new bounds are the result of a variety of methods involving filters, extension, semi-polycirculant graphs, locally linear graphs, and the modification of special graphs. Most of our bounds are from the semi-polycirculant graph generator, showcasing its efficacy for finding witness Folkman graphs.

A sequence $D = \{d_1,...d_n\}$ is a feasible degree sequence if there is a graph on $\{1,...,n\}$ such that $i$ has degree $d_i$. For such a sequence, $G(D)$ is a graph chosen uniformly at random from those with the given degree sequence. We consider sequences $\{D_\ell\}_{\ell \geq 1}$ of feasible degree sequences which have a giant component. We show that with high probability this giant component is unique, and bound its diameter and the mixing time of the random walk on it. We also bound the size and diameter of the other components and show that many of these bounds are tight.

Hall's theorem on differences of bijections characterizes the multisets $$ \{a_1,\ldots,a_{|G|}\} $$ in a finite abelian group $G$ that can be written in the form $$ a_i=b_i-c_i, $$ where both $b_1,\ldots,b_{|G|}$ and $c_1,\ldots,c_{|G|}$ are enumerations of $G$. The necessary and sufficient condition is the zero-sum condition $$ a_1+\cdots+a_{|G|}=0. $$ This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset $A$ of cardinality $|G|$ can be represented as $$ A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, $$ where $b$ and $c$ are bijections onto $G$. Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of $A$ is trivial in $ G_{\rm ab}. $ We show that this condition is not sufficient in general, even when the elements of $A$ admit an ordering whose product is the identity in $G$. The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of $A$ into product-one words whose partial-product sets tile $G$ by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in $ S_3, $ and we extend the same obstruction to infinitely many finite nonabelian groups.

We analyze a twisted Bose-Einstein Markov chain on $[k]^n$ arising from the Diaconis-Zhong twisted Burnside process. We derive explicit formulas for the transition kernel and stationary distribution of both the original chain and its lumped process, and we identify this chain as a special case of Crane's cut-and-paste process. We establish bounds on the mixing time and determine the second-largest eigenvalue of the original chain and its lumped process. Complementing the fixed-$k$ cutoff theorem of Crane and Lalley, our bounds show that both chains mix in $\Theta(\log n)$ steps when $k$ grows as a fixed positive power of $n$. For the non-twisted Bose-Einstein Markov chain, this resolves a conjecture of Diaconis. As an application, we study the Burnside processes on parking functions and labeled Dyck paths, which give novel Markov chain Monte Carlo algorithms for sampling an increasing parking function and a Dyck path, respectively, approximately uniformly at random. We show that these chains are rapidly mixing, with mixing times of $\Theta(\log n)$.

The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\det\left[\left[\left\lfloor\frac{aj-(a+1)k}n\right\rfloor\right]_q\right]_{1\leqslant j,k\leqslant n}=-\left(\frac{a(a+1)}n\right)q^{(1-3n)/2}$$ and $$\det\left[\left[\left\lceil\frac{(a+1)j-ak}n\right\rceil\right]_q\right]_{1\leqslant j,k\leqslant n}=\left(\frac{a(a+1)}n\right)q^{(n-1)/2},$$ where $(\frac{\cdot}n)$ denotes the Jacobi symbol.

Given a polyhedron (planar, $3$-connected graph) $G$, we investigate its common neighbourhood graph con($G$). For cubic ($3$-regular) polyhedra, we show that the planarity of con($G$) depends on the number of odd faces of $G$, and on their adjacency. We then prove that for all other polyhedra, con($G$) is non-planar. We introduce a novel concept for polyhedra (and more generally, for plane graphs) $G$, namely the `facial common neighbourhood graph' facecon($G$). Its definition takes into account pairs of vertices with a common neighbour on the same face of $G$. It is a spanning subgraph of con($G$), that coincides with con($G$) for cubic polyhedra. It also generalises the reverse construction of the radial graph. As part of our investigation, we also prove a technical result of independent interest: if a maximal planar graph (triangulation of the sphere) has exactly two vertices of odd degree, then they are not adjacent. We also answer several questions in extremal graph theory. Fixing the number of vertices, we characterise the polyhedra $G$ such that con($G$) is planar and the number of edges in con($G$) is minimal/maximal. We address the same problem for facecon($G$), and prove that if it is maximal planar, then $G$ has no face of length greater than $6$. We notably characterise and explicitly construct all polyhedra $G$ of maximal face length $4$ such that facecon($G$) is maximal planar.

Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg–Sárközy theorem for square differences by proving an “arithmetic level-d” inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials $h\in\mathbb{Z}[x]$. In particular, let \[ D(h(\mathbb{N}),X):= \max{|A|:\ A\subseteq [1,X]\cap\mathbb{N} \text{and}\ (A-A)\cap h(\mathbb{N})\subseteq\{0\}}. \] We prove that for every $0<\mu<1/2$ there are constants $c_0, X_{\text{min}}>0$ depending on $h$ and $\mu$ such that for every $X>X_{\text{min}}$, \[D(h(\mathbb{N}), X)\leq Xe^{-c_0(\log X)^\mu}.\] This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range $0<\mu<1/2$, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the “random sparsification” procedure of Green and Sawhney.

We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. For locally finite graphings, we define a weak isomorphism to be an edge-measure-preserving Borel bijection that preserves cycles and hyperfinite subgraphs, modulo null sets. Our rigidity theorem shows that every weak isomorphism of a weakly $3$-connected, infinitely-ended graphing is induced by an isomorphism of graphings. To our knowledge, this is the first general sufficient condition in measurable combinatorics for two given graphings to be isomorphic. We then prove a full measurable Whitney theorem, namely that every weak isomorphism between graphings is implemented by a finite composition of locally finite sequences of six measurable Whitney operations. The proofs combine measurable-combinatorial techniques, applications of spanning forests from percolation theory, and graph decomposition arguments. A central ingredient is the construction and analysis of infinitely-ended leafless subforests. With these tools we also show that, between weakly $2$-connected, infinitely-ended graphings, preservation of hyperfiniteness alone forces preservation of cycles, which is of independent interest. This work further develops the limit theory of matroids recently initiated by Lovász and provides its first application in finite graph theory. We show that Whitney's rigidity theorem is robust under sublinear errors for sequences of finite graphs with uniformly bounded degrees and suitable connectivity.

A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices $A=\{(u_i,v_i), i\in[k]\}$ and $B=\{(u_i',v_i'), i\in[l]\}$, we consider the orientations $\alpha$ of $G$ such that adding the elements of $A$ and $B$ as additional directed edges to $\alpha$ gives an orientation $\alpha'$ in which $v_i$ cannot reach $u_i$ for all $i\in[k]$, but $v_i'$ can reach $u_i'$ for all $i\in[l]$. We show that this set of orientations is equinumerous to a set of subgraphs satisfying the “same" connectivity constraints defined in terms of $A$ and $B$. We also extend our results to the enumeration of equivalence classes of orientations satisfying such connectivity constraints. Precisely, we consider the equivalence classes under cycle reversal, cocycle reversal, or cycle-cocycle reversal. We show that the equivalences classes are equinumerous to some sets of subgraphs defined by connectivity and acyclicity constraints.

We study the Temporal Exploration problem, where an agent must visit all vertices of a temporal graph while traversing at most one available edge per time step. Unlike static graphs, which can be explored in linear time, temporal constraints can substantially increase exploration time even when every snapshot of the graph is connected. To better understand the source of this complexity, we focus on a near-static setting and consider always-connected $k$-edge-deficient temporal graphs, in which each snapshot is connected and differs from a fixed underlying $n$-vertex graph by at most $k$ edges. Although such graphs are structurally close to static graphs, they can still exhibit non-trivial temporal behaviour. Prior work showed that these graphs can be explored in $O(kn \log n)$ time steps and established a lower bound of $\Omega(n \log k)$, leaving open whether linear-time exploration in $n$ is possible. We resolve this question by showing that any always-connected $k$-edge-deficient temporal graph admits an exploration schedule of length $O(nk \log k)$. Moreover, given such a temporal graph, the corresponding exploration schedule can be computed in polynomial time. The obtained bound is linear in the number of vertices up to a factor depending only on $k$, removes the extraneous logarithmic dependence on $n$, and is nearly optimal. In particular, for constant $k$, our result yields an order-optimal $\Theta(n)$ exploration time, showing that temporal exploration in this near-static regime essentially retains the linear-time character of static graph traversal.

Multilinear representability extends classical linear representability of matroids by assigning subspaces, rather than vectors, to ground elements. This notion is closely related to almost affine codes. In this paper, we introduce and study a $q$-analogue of multilinear representability for $q$-matroids, motivated by known connections between $q$-matroids, classical matroids, and rank-metric codes. We define $m$-multilinear representability in terms of almost affine matrix rank-metric codes satisfying a natural divisibility condition. We prove that nontrivial uniform $q$-matroids admit no purely multilinear representations, and we derive necessary conditions for multilinear representations of almost uniform $q$-matroids. We further show that the non-Pappus $q$-matroid, if multilinearly representable, must have block size at least $9$. Finally, we prove that no rank-$2$ $q$-matroid on $\mathbb{F}_2^4$ admits a purely $m$-multilinear representation for $1<m<4$, and we classify pure multilinearity for all $q$-matroids on $\mathbb{F}_2^3$ and $\mathbb{F}_2^4$ in the corresponding ranges. At present, no example is known of a purely multilinear $q$-matroid.

Let $D=(V(D), A(D))$ be a digraph of order $n$ and let $S\subseteq V(D)$ with $2\leq |S|\leq n$. A directed cycle $C$ of $D$ is called a directed $S$-Steiner cycle (or, an $S$-cycle for short) if $S\subseteq V(C)$. Steiner cycles have applications in reliable designs for telecommunication and transportation networks. Two $S$-cycles are called arc-disjoint if they have no common arcs. We use $\lambda_{S}^{c}(D)$ to denote the maximum number of pairwise arc-disjoint $S$-cycles in $D$. The directed cycle $k$-arc-connectivity of $D$ is defined as $$\lambda_{k}^{c} (D)=\min\left \{ \lambda _{S}^{c}(D)\mid S\subseteq V(D),\left | S \right | =k,2\le k\le n \right \}.$$ In this paper, we determine the complexity for $\lambda_{S}^{c} (D)$ on Eulerian digraphs, planar digraphs and symmetric digraphs. We also obtain exact values of $\lambda_{k}^{c} (D)$ on complete digraphs, complete bipartite digraphs and complete regular multipartite digraphs.

We investigate the multicritical scaling limit of the shifted Schur measures. Under an appropriate scaling limit and specific conditions on the continuous parameters, we explicitly determine the limit shape of strict partitions distributed according to the shifted Schur measure. We then show that, under a multicritical condition, the edge scaling limit of the correlation function converges to a determinant of the higher-order Airy kernel. This rigorously demonstrates a transition from a Pfaffian point process to a determinantal distribution in the scaling limit.

We prove that if $\Gamma$ is a connected graph with minimum degree $\delta$ and Laplacian eigenvalues $0=\mu_1<\mu_2\leqslant \cdots \leqslant \mu_n$, then the toughness of $\Gamma$ is bounded below by $\mu_2/(\mu_n-\delta)$.

We prove a lower bound on the number of directions determined by Cartesian products $A\times A$ in the affine plane over the finite field $\mathbb F_{p^2}$. Our lower bound holds for sets of size $p^{2/3}<|A|<p$, which are not contained in any affine copy of $\mathbb F_p$. The proof combines a structural result of Li and Roche-Newton on the set of directions formed by Cartesian products with a lower bound of Fancsali, Sziklai and Takáts. A key step shows that, unless the set of directions exhibits closure properties forcing subfield structure, one obtains a direction for which an algebraic multiplicity parameter in the latter theorem can be made explicit.

Temporal networks – sequences of time-stamped contacts among nodes – constitute the finest-grained representation of dynamic interaction data; however, geometric and topological analyses of such networks have remained largely confined to time-aggregated or snapshot-based approximations. Such reductions destroy the temporal ordering and interevent statistics essential for understanding spreading dynamics, synchronization, and information flow. This study proposes a geometric framework that lifts a contact-sequence temporal network into a genuine simplicial complex through a prism construction adapted from algebraic topology. On this spatiotemporal prism complex, we develop the Forman–Ricci curvature in its original CW-complex form and contrast it with an augmented variant widely used in network science. We prove that the two variants coincide under uniform weights, derive a closed-form expression for their pointwise discrepancy in the general case, and identify the precise conditions under which they diverge – conditions generically satisfied in temporal networks because temporal edges carry interval-dependent weights. Numerical experiments on three synthetic contact-network models (Erd\H os–Rényi, activity-driven, and bursty) and on the SocioPatterns Hypertext 2009 face-to-face contact dataset quantitatively confirm the theoretical predictions: the two Forman variants disagree on $56$–$67\%$ of the $1$-simplices – predominantly the temporal and diagonal simplices – while remaining strongly correlated according to the Pearson coefficient. The proposed framework provides a principled, parameter-free method for assigning discrete Ricci curvature to each contact event, thereby opening a new geometric avenue for temporal data analysis.

We determine all permutation polynomials among several families of polynomials over $\mathbb{F}_{q^3}$ for arbitrary prime powers $q$. We obtain some new families of permutation polynomials over $\mathbb{F}_{q^3}$ with simple coefficients for infinitely many characteristics. As a specific consequence, our results resolve the generalization of conjectures of Zhang, Zheng, Wang, Peng, and Li in the even characteristic. Our proofs are conceptually short and involve no complicated computations, in contrast to the proofs of results on permutation polynomials which were published previously. Moreover, we develop a totally new systematic method in this paper for the study of permutation polynomials.

We give three explicit quantum Latin squares of order $6$, with cardinalities $13$, $15$, and $17$. Throughout, vectors differing only by a global phase are counted as identical. The cardinality-$13$ construction is based on an orthogonal direct-sum decomposition $\C^6=\C^4\oplus\C^2$. The cardinality-$15$ and cardinality-$17$ constructions are based on two-dimensional Hadamard pairs supported on coordinate planes.

We derive explicit upper bounds for the number of nondegenerate critical points of a $k$-component Gaussian mixture density in $\mathbb{R}^d$, and the number of modes when the modal set is finite, together with lower bounds. By normalizing the critical-point equations by a reference component, for $k\ge2$ we get the direct Pfaffian bound \[ U_{\mathrm{het}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+2\min(d,k-1)+1\right)^{k-1}. \] For the same parameter range, an exact elimination augmented by an algebraic reciprocal variable gives the alternative bound \[ U_{\mathrm{aug}}(d,k)= 2^{\binom{k-1}{2}}(d+1)\left((2k-1)d+2k-1\right)^{k-1}. \] Thus, for $k\ge2$, the best critical-point bound is their minimum. A Morse-theoretic argument improves the corresponding finite-mode upper bound to \[ \left\lfloor \frac{\min\{U_{\mathrm{het}}(d,k),U_{\mathrm{aug}}(d,k)\}+1}{2}\right\rfloor. \] In the homoscedastic case, for $k\ge2$, the direct bound improves to \[ U_{\mathrm{hom}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+\min(d,k-1)+1\right)^{k-1}, \] an affine-rank reduction replaces $d$ by the affine rank of the component means, and an augmented homoscedastic reduction gives the dimension-free bound \[ U_{\mathrm{aug,hom}}(k)=2^{\binom{k-1}{2}+1}(2k)^{k-1}. \] On the lower-bound side, for $d,k\ge 2$ we obtain \[ L_{\mathrm{bin}}(d,k)=k+\max_{2\le r\le \min(d,k)}\binom{k}{r}, \] together with a padding-product family that in particular implies the linear lower bound $d+k-1$, and a seed-closure principle that packages product and padding constructions. We further give explicit bounds for the number of connected components of the critical set.

For the OEIS sequence A002627, defined by the inhomogeneous first-order recurrence $a(n) = n\,a(n-1) + 1$ with $a(0) = 0$, R.~J.~Mathar recorded in February 2014 the conjectured second-order homogeneous recurrence \[ a(n) - (n+1)\,a(n-1) + (n-1)\,a(n-2) = 0, \qquad n \ge 2, \] which has remained marked as a conjecture on the OEIS for over a decade. We give three short proofs. The first is two lines: subtract the defining recurrence at adjacent indices and the constant cancels (we call this homogenisation). The second reads off the same relation from the exponential generating function $F(x) = (e^x-1)/(1-x)$. The third is a Pascal-rule telescoping on the binomial-sum form $a(m) = \sum_{k=0}^{m-1} k!\binom{m}{k}$. All three derivations are elementary, requiring nothing beyond undergraduate techniques. We remark that the same homogenisation trick clears an entire class of “Conjecture: \dots” entries on the OEIS, namely sequences satisfying $a(n) = p(n)\,a(n-1) + q(n)$ with simple $q$.