Papers

We prove stronger forms of two conjectures of Berkovich and Dhar concerning a coefficientwise inequality and pairs of hooks of length two in bounded partitions. The coefficientwise inequality implies the corresponding inequality for hook pairs. In particular, for each fixed size and common bound on the largest part, the total number of such pairs in odd partitions is at least that in distinct partitions. We give combinatorial proofs by explicit weight-preserving injections based on Glaisher's bijection and analytic proofs using finite product identities and known hook generating functions.

This is an essentially derivative note, whose goal is to interpret the 'whole-cube' bound of OpenAI for binary codes in a possibly somewhat more accessible way. This is attained, in part, by connecting it to some previously known results. All of the heavy technical lifting in the interpretation below is also due to the OpenAI language models ChatGPT 5.6 and ChatGPT 6. With that, we hope that the statements of the main results, the ensuing discussion, and the arguments themselves may be of some interest.

Improper edges were introduced by Shor to refine Cayley's formula for rooted labeled trees. Zeng established a connection between Shor's refinement and the Ramanujan polynomials. Let $\mathscr{T}_n$ denote the set of rooted labeled trees on $[n]=\{1,\ldots,n\}$. We prove that the number of improper edges in a uniformly random tree in $\mathscr{T}_n$ is asymptotically normal as $n\to\infty$, with mean and variance asymptotic to $\mu n$ and $\sigma^2 n$, respectively, where $\mu=e-2$ and $\sigma^2=e^2-3e+1$. This phenomenon was observed by Chen, and the proof presented here was developed through human–AI collaboration.

For integers $n\ge3$ and $2\le\Delta\le n-1$, let $\lambda_1(n,\Delta)$ be the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $\Delta$. Liu conjectured that for each fixed integer $\Delta\ge3$, \[ \lim_{n\to\infty}n^2\bigl(\Delta-\lambda_1(n,\Delta)\bigr)= \begin{cases} (\Delta-1)\pi^2/4,&\text{if $\Delta$ is odd};\\ (\Delta-2)\pi^2/2,&\text{if $\Delta$ is even}. \end{cases} \] He proved the case $\Delta=3$ and $\Delta=4$. We prove the conjecture when $\Delta\ge5$ is odd.

Let $\Ke$ be obtained from $K_7$ by deleting two independent edges. We prove that every 5-connected graph on $n\ge7$ vertices with at least $4n-9$ edges contains a $\Ke$ minor, settling Conjecture~1.4 of Dvo\v rák, Norin and Rahman (arXiv preprint 2609.17760v1). The bound is sharp. We prove the stronger statement that every $4$-bilight graph on $n\ge4$ vertices with at least $4n-9$ edges contains either a $\Ke$ minor or a $K_6$ subgraph. Within their reduction framework, we strengthen the rooted-minor theorem. We show that every $4$-light 5-rooted graph of rooted $4$-density at least two has a model with two nonroot vertices and at most one missing edge incident with them. At the critical density, reductions preserve density exactly, which prevents them from creating a new $K_6$ subgraph.

For $k\geq2$, let $\alpha_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $\beta_k$ be the supremum of the $c<1$ for which, with probability tending to one as $D\to\infty$, the set $\mathbf A\cap[D^c,D]$ has $k$ distinct subsets with the same sum. We prove that $\alpha_k=\beta_k/(1-\beta_k)$, resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027–1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.

It is known that the inverse of an invertible real square matrix satisfying the treeangle property, is a treediagonal matrix. A converse statement also holds. We show that the verbatim analogues are not true for the Moore-Penrose inverse, and obtain the precise structure of graphs corresponding to the Moore-Penrose inverse of matrices possessing the treeangle property.

For $n\ge6$, $m=\binom n2$, let the complete-graph incidence family on $K_n$ have the vertices of $K_n$ as columns, its edges as rows, and the incidence graph as one-edge graph. The universal cell bound of Löfberg and Qi gives $z_2(m,n)\le Z(n):=\lfloor n(n-1)(n+2)/4\rfloor$. We implement the nested one-factorization construction of that family and determine exactly what it certifies. For $n=2q$ with $q$ an odd prime, $q\ge5$, the construction has no hole and the cross-factor transfer equations apply verbatim, giving $z_2=z_{SL}=z_{RL}=Z(n)$; $n=6$ is settled by a separate cyclic witness. For odd $n=2p+1$ the near-perfect one-factorization yields Hamilton paths closed into odd cycles, and the transfer argument breaks: that scheme certifies only $R(G_p)\le Z(n)$. The orders $n=7,8,9,12,13,16,17,18,20,21$ are settled by explicit configurations of a different shape, each attaining the cell bound and satisfying $(\mathrm{RW}3^+)$, obtained from a larger configuration by deleting vertex stars and repairing the restricted grid. At each of these orders $z_2=z_{SL}=z_{RL}=Z(n)$; all remaining orders are conjectural. The first settled order $n=7$ is the only one whose grid has a hole, grounded by a zero-companion rule. Machine-readable configurations and a certificate checker accompany the paper.

All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2^{-\Omega(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integral graphs rely on Cayley graphs over Abelian groups. In this article, we introduce a new family of integral graphs obtained from the groups. The construction of our graphs from groups is different from the construction of Cayley graphs. A spectral uniqueness theorem is established, which shows that each member of the infinite family is determined by its adjacency spectrum among all finite simple graphs. We also present recursive constructions that generates larger members of the family from smaller ones, providing a scalable class of integral graphs. Finally, we investigate algebraic realizations of these graphs as complements of Proper Prime Order Element Graphs of finite $2$-groups and obtain conditions characterizing such realizations. We also observe that the graphs obtained from different non-isomorphic groups have cospectral graphs.

We propose a bounded degree SOS+SONC hierarchy for constrained polynomial optimization, termed B-SOS+SONC. Starting from Lasserre's bounded-degree SOS framework, we enlarge the certificate cone from SOS to the recently introduced SOS+SONC cone, thereby combining the algebraic strength of semidefinite relaxations with the sparse structure captured by circuit polynomials. We show that, for each fixed certificate degree, the resulting hierarchy is complete, that is, its optimal values are monotone and converge to the global optimum. Moreover, we derive an explicit SDP-REP reformulation, so that each relaxation can be solved within a tractable convex optimization framework over semidefinite and relative entropy cones. Beyond the optimization hierarchy itself, we investigate structural properties of the SONC cone and introduce the notions of first-order and second-order SONC-convexity. This leads to a new sufficient condition for first-level exactness of the B-SOS+SONC hierarchy. Numerical experiments illustrate that the proposed hierarchy often yields tighter lower bounds than the B-SOS relaxation while remaining tractable.

The Ehrhart series of the $n$th Birkhoff polytope is $\sum_{r\geq0}H_n(r)z^r$, where $H_n(r)$ counts nonnegative integer $n\times n$ matrices whose row and column sums all equal $r$. We present an exact method for computing this series using constant terms and finite fields. A root filter expresses $H_n(r)$ as a weighted sum of the values $h_r(M)^n$, where $h_r$ is the complete homogeneous symmetric polynomial and $M$ ranges over multisets of $m$th roots of unity with $m=r+1$. Constant-term cancellation reduces the evaluation of $h_r(M)$ to a sum over repeated elements $a$ of $M$. For a particular $a$ of multiplicity $\mu_a$, the computation uses a generalized Todd coefficient of degree $\mu_a-2$. Sums of $h_r(M)^n$ over selected multiplicity classes are handled using symmetric function techniques. Together with the remaining individual evaluations, this gives $O_n(m^{n-5}+m^4)$ field operations for each admissible prime and fixed $n\geq5$. An explicit bound and the Chinese remainder theorem recover the integer counts, and Ehrhart symmetry determines the full series. The same method applies to the World Cup problem, which counts the same matrices with diagonal entries required to be $0$. We prove correctness and compute complete series for both families through order $12$. The Birkhoff series for orders $10$–$12$ and the World Cup series for orders $9$–$12$ are tabulated in the appendices.

Let $\tau(G)$ denote the maximum number of edge-disjoint spanning trees in a connected graph $G$ of order $n$, and let $\rho_D(G)$ denote its distance spectral radius. For an integer $k\ge2$, Fan, He and Zhao [Discrete Appl. Math. 376 (2025) 31–40] obtained a sharp distance spectral radius condition for $\tau(G)\ge k$ when $n\ge2k+6$. In this paper, we fill the gap $2k\le n\le2k+5$ and thus complete the result for all $n\ge2k$. The extremal graph given by Fan, He and Zhao remains valid for $n\ge2k+2$, while we determine the unique extremal graph for each of the orders $n=2k$ and $n=2k+1$. We further obtain sharp distance spectral radius conditions and characterize all extremal graphs under the minimum degree condition $\delta(G)\ge k$ for all $n\ge2k$. Finally, for graphs with the stronger minimum degree condition $\delta(G)\ge6k-4$ and order $n\ge2\delta(G)+2$, we obtain a sharp distance spectral radius condition ensuring $\tau(G)\ge k$ and determine the unique extremal graph.

We study the zero-run structure of the $(3,2)$-Raney numbers modulo a prime $p$. For every prime $p$, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in $0\le n<p^m$. Interestingly, these results fall into three cases: $p=2$, $p=3$, and $p\geq5$, and the behaviors in these three cases are very different. For $p=2$, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For $p=3$, we characterize exactly the nonzero terms and determine their residues. For $p\ge 5$, the zero runs are governed by a multiscale system of residue intervals modulo powers of $p$, from which both the record values and the complete zero-run spectrum are obtained.

Let $\nu_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+\nu_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};q^{2^{r+1}r}\bigr)_\infty$. Our method is to regard the exponent map $m\mapsto m(\nu_2(m)+2)$ as a dynamical system on the positive integers: the factors of the associated quotient link into chains along its forward orbits, and the three orbits seeded at $1$, $4$ and $5$ are pairwise disjoint and telescope to exactly $1/\bigl((1-q)(1-q^4)(1-q^5)\bigr)$. The exponent triple $(1,4,5)$ is admissible in the sense of earlier work by Liu, which yields the desired factorization into two series with nonnegative coefficients. This orbit telescoping technique appears to be a mechanism complementary to the Pólya–Szeg\H o criterion that underlies most existing positivity results of this kind in the literature.

We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of $H$. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.

The first author and Dong \cite{DD} proposed three conjectures on antichain generating polynomials. Jiang \cite{Jiang} recently proved Conjectures 4.3 and 4.5, concerning real-rootedness and \(\gamma\)-positivity. We prove Conjecture 4.2 by adapting his method from \([k]\times [2] \times [n]\) to \([k]\times P_{n,s} \), where \(P_{n,s}\) is a two-row Ferrers shape. We establish real stability for a family of bivariate polynomials associated with adjacent shapes, then apply the Chudnovsky-Seymour compatibility criterion to obtain a common interleaver. As noted in \cite{DD}, Conjecture 4.2 also implies Conjecture 4.3.

Let $A\subset\mathbb{R}$ be a finite set. Combining the multiplicative slope estimate of Rudnev–Stevens, Cushman's higher-energy regularization, Shakan's $d^+$–$d^\times$ decomposition, and Solymosi's classical sum–product estimate, we prove \[ |AA|^{204}|A+A|^{301}\gtrsim |A|^{675}, \] where $\gtrsim$ suppresses a fixed polylogarithmic factor in $|A|$. Consequently, for every $\varepsilon>0$, \[ \max\{|A+A|,|AA|\}\gg_\varepsilon |A|^{135/101-\varepsilon}. \] The proof is organized around two intermediate estimates. For every nonempty finite set $B\subset\mathbb{R}_{>0}$, \[ d^\times(B)|BB|^{12}|B+B|^{16}\gtrsim |B|^{38}, \] whereas for every nonempty finite set $U\subset\mathbb{R}$, \[ d^+(U)^{17}|U+U|^{29}\gtrsim |U|^{46}. \]

We prove a colorful quantitative Helly theorem for volume with the optimal number $2d$ of colors. If every rainbow intersection from $2d$ finite families of convex sets in $\R^d$ has volume at least one, then the intersection of one family has volume at least $d^{-O(d^2)}$. We also prove a colorful quantitative Steinitz theorem for origin-centered ellipsoids of different shapes. The proof uses a common normalization of positive operators and a lift that produces two rainbow bases with large determinants.

Let $n$ be a positive integer. A translation-invariant total order (TITO) with period $n$ is a total order of the integers that is invariant under translations by multiples of $n$. These structures arise naturally in the study of Coxeter groups. In particular, real $n$-TITOs are in bijection with biclosed sets of positive roots of the affine symmetric group $\widetilde S_n$. Barkley and Defant recently introduced pattern avoidance for TITOs and used it to define the affine Tamari lattice. The enumeration of TITOs avoiding a single pattern of length $3$ is due to Crites and Barkley–Defant. We extend this work to TITOs avoiding two patterns. Our main results include a complete enumeration of TITOs that avoid a pair of patterns in $S_3\times S_3$, as well as of TITOs that avoid a pair $(p, q)$ with $p \in S_3 \setminus \{123, 321\}$ and $q \in S_4$. Furthermore, we provide an explicit construction of the inverse of the bijection between $312$-avoiding TITOs and noncrossing arc diagrams, thereby extending the combinatorial framework introduced by Barkley.

We construct an explicit quaternary Legendre pair of length $64$. To the best of our knowledge, this is the first reported construction at this length.

We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.

Motivated by Eliahou's 64-modular Hadamard construction at the smallest open Hadamard order n=668, we introduce an explicit half-flip family of 32-modular Hadamard matrices at orders n=4L for L = 3 (mod 8). A master identity reduces the four-sequence Golay-quadruple condition under the half-flip ansatz (s, s*, sq, (sq)*) to a single-sequence type-restricted autocorrelation c_k^tau(s). The construction yields a closed-form expression for c_k^tau, a true Hadamard matrix at L=11, and 32-modular matrices at every L = 3 (mod 8), including the open orders n=716 and n=1132. Existence of 32-modular at L = 3 (mod 4) is due to Eliahou-Kervaire (2001); Eliahou's 2026 follow-up in J. Algebraic Combin. gives 64-modular matrices at L = 3 (mod 16) and L = 7 (mod 32) via the same ansatz and correlation identity we call the master identity, subsuming our construction on that residue subclass. Our contribution is therefore primarily structural. Applying Barrera Acevedo-O Cathain-Dietrich (2019) and Alvarez et al. (2020), the family is non-cocyclic over any group at every prime L in {11, 19, 59} in the YES set, yet pseudococyclic over the Goethals-Seidel Moufang loop GS_{4L} at every L. A half-flip H-set decomposition theorem parameterizes the symmetric difference of any two family elements by a single sequence flip set, giving the family the structure of a length-L Hamming cube. A symbolic mod-tower verifier (mod-8 is F_2-linear) classifies true Hadamards in the family through k=14 (L <= 115): the YES set is empirically bounded by k=7, refuting four H4 predictions and excluding L in {179, 283} within the ansatz. A Grobner basis at L=11 exhibits a previously unrecorded even-T0-block linear identity. All code and JSON certificates: github.com/michelkulhandjian/hadamard-halfflip-structural

Let $G$ be a connected graph. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertex set consists of all subsets of $k$ vertices of $G$, where two of them are adjacent whenever their symmetric difference is an edge of $G$. Every automorphism of $G$ induces one of $F_k(G)$, as does complementation when $k=|G|/2$; automorphisms of this form are called induced. Fabila-Monroy et al.\ (Graphs and Combinatorics 42, 2026) show that token graphs can have many non-induced automorphisms, arising from twin cuts. These are cut sets $\{x,y\}$ whose two vertices have the same neighbours (other than themselves) in $G$. These non-induced automorphisms send configurations with a prescribed number of tokens on each component of $G\setminus \{x,y\}$, totalling $k-1$, and exactly one token on one vertex of $\{x,y\}$, to the configuration obtained by moving (flipping) the token at $\{x,y\}$ to the other vertex of $\{x,y\}$. These automorphisms send an induced $4$-cycle generated by moving two tokens on two disjoint edges to one generated by moving two tokens on a $4$-cycle; thus introducing what we call a twist. We prove a partial converse: if an isomorphism $\varphi\colon F_k(G)\to F_{k'}(G')$ sends some $4$-cycle generated by moving two tokens on two disjoint edges to one generated by moving two tokens on a $4$-cycle, then $G$ and $G'$ have twin cuts $\{x,y\}$ and $\{x',y'\}$, respectively. We also show that any twist can be undone by composing $\varphi$ with twin-cut flips.

Let $G=(G_1,...,G_n)$ be a graph system on a common $n$-vertex set $V$. For $v\in V$, set \[ \delta(v)=\min_{i\in[n]}d_{G_i}(v), \qquad E_{\cap}=\bigcap_{i=1}^n E(G_i), \] and let $d^c(u,v)$ denote the length of a shortest rainbow $u$–$v$ path. We study Hamiltonicity conditions imposed only on pairs at small rainbow distance. Suppose that every $G_i$ is $2$-connected and that \[ \delta(u),\delta(v)<\frac n2, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap}. \] Our main result shows that any system satisfying this local Fan-type condition but containing no rainbow Hamilton cycle must satisfy \[ E_{\cap}=\varnothing, \qquad \sum_{v\in V}\delta(v)\le\frac{n^2-1}{2}. \] Consequently, either $E_{\cap}\ne\varnothing$ or $\sum_{v\in V}\delta(v)\ge\lceil n^2/2\rceil$ guarantees a rainbow Hamilton cycle. We further prove that the local Ore-type condition \[ \delta(u)+\delta(v)<n, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap} \] already forces a rainbow Hamilton cycle. Along the way, the Fan-type hypothesis yields a rainbow Hamilton path and a rainbow $(n-1)$-cycle. For diagonal systems, our conditions reduce to the classical Fan and local Ore conditions. The proofs combine rainbow rotation–extension with a double-counting argument in an auxiliary digraph. Keywords: graph system; rainbow Hamilton cycle; rainbow distance; Fan-type condition; Ore-type condition; rotation–extension.

Let $\mathbb{R}\langle X,Y\rangle$ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone $\Theta$ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space $Z$ of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on $Z$ they coincide: for $f\in Z$, trace-positivity is equivalent to membership in $\Theta_3$, to non-negativity of the trace on pairs of $2\times2$ symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in $\Theta$ reduces to a $4\times4$ positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-$(4,2)$ and $(2,4)$ parts of $f$; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-$(3,3)$ part is then confined to a rectangle whose sides are the geometric means $\sqrt{p_+q_+}$ and $\sqrt{p_-q_-}$ of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of $2\times2$ symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of – a trace-positive polynomial outside $\Theta$ – arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of $f$ is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside $\Theta$.

In 1984, Frank Harary introduced the first convexity games in graphs, all of them based on the geodesic convexity, which is the graph convexity related to shortest paths. In 2024, Araújo et al. obtained the first PSPACE-hardness proofs on some of these geodesic games and generalized them to any graph convexity. In this paper, we investigate convexity games on several known induced path convexities: the monophonic $\mathrm{m}$-convexity and the $\ell_k$-convexities, based on induced paths and on induced paths of size at most $k$. We prove that the hull games $\mathrm{CHG}_{\mathrm{m}}$, $\mathrm{CHG}_{\ell_k}$ and their misère variants are PSPACE-complete for every $k\ge2$ even in graphs with diameter at most 3. We also use the Sprague-Grundy Theory to obtain a polynomial time algorithm to decide the winner of the games $\mathrm{CHG}_{\mathrm{m}}$ and $\mathrm{CHG}_{\ell_k}$ for any $k\ge2$ in disjoint unions of paths and cycles. For $k\ge3$ odd, we prove that Alice (1st player) wins $\mathrm{CHG}_{\ell_k}$ in the path $P_n$ if and only if $n$ is odd and she wins in the cycle $C_n$ if and only if $n=3$ or $n=\alpha\cdot (k+1)-1$ with $\alpha\ge2$. For $k\ge2$ even, the only periodic nimber sequences of $\mathrm{CHG}_{\ell_k}$ obtained through extensive computational testing occurred for $k=2^h-4$ with $h\ge3$, e.g, $k\in\{4,12,28,60,\ldots\}$. In this case ($k=2^h-4$ with $h\ge3$), we prove that the nimber sequences of $\mathrm{CHG}_{\ell_k}$ in $P_n$ and in $C_n$ are periodic and Alice loses (resp. wins) in $P_n$ (resp. $C_n$) with $n>k$ only when $n=3k+4$ (resp. $n\in\{2k+1,5k+3\}$). Finally, we show that, for $k=2$, the game $\mathrm{CHG}_{\ell_2}$ in paths $P_n$ is closely related to the classical game Couples-are-Forever of J. H. Conway: it is still an open problem if the nimber sequence is periodic or not and Alice loses only for 12 values of $n$ up to $50$ million.

For a $3$-uniform hypergraph $H$, let $\sigma(H)$ be the minimum over the vertices of $H$ of the spectral radius of the link. Lin, Lu, Yuan and Zhao conjectured that $\sigma(H)>\tfrac{2n}3-2$ forces a perfect matching in a $3$-graph of order $n$ divisible by three, and Lu and Yuan conjectured that $\sigma(H)>\tau(q)$ forces one in a $q$-balanced $3$-partite $3$-graph, where $\tau(q)=\sqrt{q(q-1)/2}$ for odd $q$ and $\tau(q)$ is given by a quartic for even $q$. In this paper, we prove both perfect matching conjectures for large order. Each proof passes through a stability theorem for fractional matchings: a fractional vertex cover of deficient weight with a zero coordinate is obtained, the link of that vertex carries the induced cover, and the spectral radius of such a link is bounded by an inequality whose extremal cases identify the corresponding space barriers.

We prove that the principal minors of an orthogonal projection matrix determine the coordinatewise absolute value of its image. More precisely, if $W_1,W_2\subset \mathbb{C}^n$ are subspaces whose orthogonal projection matrices have equal corresponding principal minors of all orders, then $|W_1|=|W_2|$, where $|W|$ is the image of $W$ under the coordinatewise absolute value map. The proof is divided into different cases related to the connectivity of the associated matroids. We provide a stratification of the Grassmannian in terms of the connectivity levels and give a structure theorem for the elements of each stratum.

For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are those whose coordinate gcd is not divisible by the $r$th power of any prime. Fix $1\le q\le \infty$. A lattice point is $R$-deep if every lattice point in the closed $\ell_q$-ball of radius $R$ centred there is non-$r$-free. We mark each finite nearest-neighbour component of non-$r$-free points containing an $R$-deep point by its lexicographically least such point. Uniformly for $1\le q\le\infty$, the densities of $R$-deep points and of these representatives both have the asymptotic form $\exp\left\{-\frac{v_{d,q}R^d}{\zeta(dr)}\left[d(dr-1)\log R+dr\log\log R-\Upsilon_{d,r,q}+r\frac{\log\log R}{\log R}-\frac{\Lambda_{d,r,q}}{\log R}+O_{d,r}\left(\frac{(\log\log R)^2}{(\log R)^2}\right)\right]\right\}$ as $R\to\infty$, where $v_{d,q}$ is the volume of the unit $\ell_q$-ball and $\Upsilon_{d,r,q},\Lambda_{d,r,q}$ are explicit constants. For deep points, this sharpens the positive-density hole constructions of Baake, Moody, and Pleasants and of Pleasants and Huck, and the latter authors' upper bounds for sparse-pattern frequencies. For $d=1$, consider the densities of $r$-free integers followed by at least $g-1$ consecutive non-$r$-free integers, or by exactly $g-1$ such integers and then another $r$-free integer. Both have the asymptotics $\exp\left\{-\frac{g}{\zeta(r)}\left[(r-1)\log g+r\log\log g-\widehat{\Upsilon}_r+r\frac{\log\log g}{\log g}-\frac{\widehat{\Lambda}_r}{\log g}+O_r\left(\frac{(\log\log g)^2}{(\log g)^2}\right)\right]\right\}$ as $g\to\infty$, where $\widehat{\Upsilon}_r=\Upsilon_{1,r,1}+(r-1)\log2$ and $\widehat{\Lambda}_r=\Lambda_{1,r,1}+r\log2$. The exact-gap expansion improves Grimmett's leading asymptotic and refines the formula in Jiang's recent preprint. For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\log R)^r)$.

We introduce back stable standard elementary monomials and their complete homogeneous analogues. These families form bases of the back stable Schubert ring and record the eventual standard elementary monomial expansions of left-stabilized Schubert polynomials. Dynkin reversal exchanges the two bases while preserving their coefficients. We also show that orbit sums of the back stable coefficients recover the elementary and complete homogeneous coefficients of Stanley symmetric functions. Finally, we study the shifted specialization polynomial $\mathcal{T}_w(t)=\mathfrak{S}_{1^t\times w}(1)$, considered by Fomin and Kirillov, which counts reduced pipe dreams of $1^t\times w$. We express this polynomial in terms of back stable standard elementary monomial coefficients.