Papers

Let $K$ be a finite pure ridge-connected $d$-dimensional simplicial complex with at least two facets, and let $m_K(\lambda)$ denote the multiplicity of a positive eigenvalue $\lambda$ of its upper Laplacian $L_{d-1}^{\mathrm{up}}(K)$. We obtain upper bounds for $m_K(\lambda)$ in terms of the basic combinatorial parameters $f_d(K)$ and $f_{d-1}(K)$, the numbers of facets and ridges, $p_d(K)$, the number of pendant facets, and $q_d(K)$, the number of quasi-pendant ridges. The quantity $\theta_d(K)=d f_d(K)-f_{d-1}(K)+1$ arises as the cyclomatic number of a natural reduced facet–ridge incidence graph. We prove \[ m_K(\lambda)\le 2\theta_d(K)+p_d(K), \qquad m_K(\lambda)\le 2\theta_d(K)+q_d(K)\quad(\lambda\ne d). \] We also characterize the complexes for which equality is attained in these bounds. Finally, for every finite pure $d$-dimensional simplicial complex, we prove the sharp inequality $m_K(d)\ge p_d(K)-q_d(K)$, which is a higher-dimensional analogue of Faria's inequality. For $d=1$, our results reduce to the corresponding multiplicity bounds for graph Laplacians.

For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erdős-Sós theorem, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1,$ $$\operatorname{ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r},$$ where $n=a(k-1)+b$ with $0\le b<k-1$. In this paper, we confirm the conjecture for $r\ge \left\lceil (2k-1)/3\right\rceil$ and characterize all extremal graphs.

Let $A$ be an $n$-element set of positive real numbers, let $FS(A)$ be its set of subset sums, and put $T_n=\binom{n+1}{2}$. For every fixed integer $C\geq-1$ and all sufficiently large $n$, we classify the sets satisfying $$ |FS(A)|\leq T_n+n+C+1. $$ Each such set is commensurable. Its unique primitive integer normalisation $B$ either satisfies $\sum B\leq T_n+n+C$ or belongs to an explicit exceptional family specified by a missing element $m\in\{1,2\}$ and an integer partition of $C+m$ or $C+m+1$. If $P$ denotes the partition function, the exceptional family has exactly $$ P(C+1)+2P(C+2)+P(C+3) $$ primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large $i$, adjoining the largest element to the preceding $i-1$ elements creates only $i+e$ new subset sums, where $e$ is bounded, then the $i$-element set is a dilation of $[1,i+e]_{\mathbb{Z}}$ with exactly $e$ elements deleted. Conversely, every such deletion pattern has increment excess $e$. The proof combines a stabiliser argument in $\mathbb{R}/x\mathbb{Z}$, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for $C\leq-2$, this completes the eventual fixed-defect classification for every integer $C$.

Anti-Latin squares were introduced in connection with non-linear secure network coding, and the extremal problem for large mutually orthogonal families is motivated by that setting. We study the maximum size $N_A(d)$ of a family of mutually orthogonal anti-Latin squares of order $d$. We prove that $N_L(d)+1\le N_A(d)\le N_L(d)+2$ for every $d\ge 3$, where $N_L(d)$ denotes the classical maximum size of a family of mutually orthogonal Latin squares of order $d$, and we show that in fact $N_A(3)=N_L(3)+1$ whereas $N_A(d)=N_L(d)+2$ for every $d\ge 4$. The upper bound is obtained by passing through balanced matrices, while the lower bound is given by a deterministic permutation argument. For all $d\ge 8$, and also for the exceptional order $d=6$, the upper bound is shown to be attainable by a general probabilistic construction. On the structural side, we show that a saturated family of size $d+1$ induces an affine plane of order $d$, and that the saturated case is characterized by the existence of an anti-coordinate grid decomposition; after transporting this condition to the fixed cell set $[d]^2$, it becomes a direction-completeness condition on the corresponding row-blocks and column-blocks. The remaining small orders are treated separately: $d=3$ is handled by direct analysis and classification of orthogonal triples, $d=4$ by an explicit saturated construction and an analysis of its finite-geometric structure, and $d=5$ and $d=7$ by explicit saturated examples arising from the random-grid framework. Thus $N_A(d)$ is determined in terms of $N_L(d)$ for every $d\ge3$, and its numerical value is obtained explicitly for every $3\le d\le9$.

Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).

I give an elementary categorical lift of the three-variable Negami polynomial and of the boundary state structure underlying the splitting formula.

We study avoidance of patterns of length $3$ with three distinct letters in canon permutations. We reduce the problem to studying pattern avoidance in lattice words and show that there are $6$ such pattern avoiding classes. This shows that there are $12$ classes for the original Canon permutation pattern avoidance problem. We also give descent refinements for these classes and classify the patterns for which the descent enumeration gives palindromic and $\gamma$-positive polynomials. When the polynomials are $\gamma$-positive, we explain the $\gamma$-positivity through a group action analogous to Foata-Strehl valley hopping. Additionally, we study the avoidance of patterns in the relabelling orbit of $1213, 12112, 1231$ after a conjecture about their cardinalities by Laudone and give bijective proofs for the results.

We introduce and study equivariant analytic spectral invariants associated with the Aç{\i}kmeşe lift of a graph with self-loops $G_S$. The canonical $\mathbb{Z}_2$-action yields an orthogonal isotypic decomposition of the lifted Laplacian into the anti-symmetric part $\mathcal{L}(G_S)$ and the symmetric part $M_{sym}$, from which we prove that the spectrum of $\mathcal{L}(G_S)$ is exactly the even-indexed spectrum of the lifted Laplacian. We express twisted moments of these blocks as traces of the generalised twisted moment operator against the corresponding $\mathbb{Z}_2$-projections, and obtain a tight upper bound under certain restriction with a characterization of the equality case. We introduce the equivariant heat character and establish a trace-norm stability estimate. For the matrix element of the resolvent $(pI+\mathcal{L}(G_S))^{-1}$ associated with the loop vector, we derive a Laplace-transform identity for the equivariant heat character, a determinant formula, a spectral representation, and a Laurent expansion. We further characterize the case where $S$ is a union of connected components through resolvent and equivariant heat-character identities, and obtain explicit equivariant heat characters for joins of full-loop graphs and the line graph of a full-loop connected regular graph. Finally, we introduce the regularized equivariant heat integral and derive spectral and trace formulas for it.

We introduce $L$-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the $L$-functions arising from unramified coverings of algebraic curves. These $L$-functions are defined using the Riemann–Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two graphs without bridges have isomorphic Jacobians and their $L$-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that Lorenzini's zeta function is invariant under contraction of bridges, explaining the necessity of the bridge-free hypothesis in the main theorem. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determine the Laplacian lattice.

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle – a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

We study the Schur-positive cone in $\Rspace_{n,\mathbb R}\coloneqq \operatorname{span}_{\mathbb R}\{p_d^{n/d}:d\mid n\}$ through its basis $Q_{n,d}\coloneqq\ell_{n/d}^{(1)}[p_d]$, where $\ell_m^{(1)}$ is the Frobenius characteristic of the representation induced to $S_m$ from a faithful linear character of the subgroup generated by an $m$-cycle; brackets denote plethysm. A Boolean $Q$-sum is a sum of distinct elements of this basis. We give a unified proof of four conjectures of Sundaram on Schur positivity by classifying all Schur-positive Boolean $Q$-sums; the case of sums over divisors up to a prescribed bound recovers Hou's theorem. Specifically, for a nonempty set $J$ of divisors of $n$, the sum $\sum_{d\in J}Q_{n,d}$ is Schur-positive exactly when $1\in J$ and, for even $n$, $n\in J$ implies $n/2\in J$. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function $\sum_{d\mid n}c_d(n/d)^2p_d^{n/d}$, where $c_d(r)$ is the Ramanujan sum, has a positive coefficient of $s_\lambda$ for every $n\ge1$ and $\lambda\vdash n$, except when $n\equiv2\pmod4$ and $\lambda=(1^n)$, in which case the coefficient is zero. We prove that an element of this space has integral Schur coefficients if and only if its $Q$-coordinates are integral. The Boolean classification also determines the convex hull of the Schur-positive Boolean points with $Q_{n,1}$-coordinate $1$. We compute its Ehrhart polynomial and volume, prove its integer decomposition property, and determine the Hilbert basis of its cone. For $n\ge18$, we prove that setting the coefficient of $s_{(n)}$ or $s_{(1^n)}$ equal to $0$ or $1$ defines a facet of the section of the Schur-positive cone with $Q_{n,1}$-coordinate $1$. The positivity results and coordinate formulas also yield inequalities for major-index residue multiplicities.

Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Zhang has given a complete proof of this conjecture by a centered pseudoinverse and spectral variance method. We present an independently obtained, structurally different proof of the strict even-dimensional inequality. Starting from the structural identities of Frankel and Urschel, we derive an exact global defect budget and combine binary rounding, fixed intersections, and Gram projection. A ten-row obstruction handles every even $n\ge66$; a finite exact calculation handles $4\le n\le64$, $n\ne6$; and a multi-column energy argument treats $n=6$. The order-two case is elementary. The even-dimensional argument is formalized in Lean 4, conditional on Frankel–Urschel Lemma 2.1 as an explicit external mathematical input. The finite evaluations use Lean's native evaluator; their trust boundary and exact certificates are documented. Together with Cheng's odd-dimensional theorem, the argument recovers the full S-matrix theorem and its equality characterization.

We develop a methodology for designing infinite products of rational blocks whose exponents are polynomials in the index $k$. Matching power sums of the slot constants through order $n$ forces the Type-$N$ product with binomial exponent $\binom{n+k-2}{n-1}$ to converge to a ratio of Vignéras multiple gamma values $\Gamma_n$; an analogue finder lifts any Type-1 evaluation to every higher type, and integer combinations of binomial exponents then realise arbitrary integer-valued polynomial exponents, yielding explicit products with exponents $k$, $k^2$, $k^3$, ... for constants such as $\pi/2$, $\sqrt{2}$, $e^{2K/\pi}$, and rational multiples of $\pi^{M!}$ (Part I). As the main application (Part II) we prove that for every positive integer $n$, a finite multiple-gamma template $\mathcal{S}_n$ with generalised Eulerian weights $T(n,k)$ (OEIS A225118) evaluates the Duke-Imamoğlu expression $\mathcal{D}_n = \beta'(-n) + (\log 4)\,\beta(-n)$. For odd $n$ this yields a convergent Wallis-Eulerian product for $e^{\beta'(-n)}$; for even $n$ the raw product diverges. The proof expands the template through the multiple-gamma functional equation, evaluates the quarter-integer coefficients in closed form, and identifies the resulting Eulerian-binomial sums with Duke's polynomials $P_{n+1,\ell}$.

Motivated by the concept of polarity introduced by G. Birkhoff for a binary relation on a set, we introduce a concept of orthogonality in a poset with $0$. A pair of operators $f$, $g$ on a poset with $0$ is called orthogonally adjoint if $f(x)$ is orthogonal to $y$ if and only if $x$ is orthogonal to $g(y)$. We characterize the existence and the uniqueness of $g$ for given $f$ and describe basic properties of orthogonal adjointness. We present constructions of orthogonally adjoint pairs in pseudocomplemented posets. If a given operator $f$ is an order-isomorphism of a pseudocomplemented poset satisfying some natural properties then the corresponding adjoint $g$ can be described explicitly. Moreover, if $f$ and $f^{-1}$ are bijective $\perp$-morphisms then they are orthogonally adjoint, too. Finally we show that a given pair of orthogonally adjoint mappings on a poset $\mathbf P$ may not be extendable to the Dedekind-McNeille completion of $\mathbf P$ and we present sufficient conditions for the existence of such an extension. We also provide sufficient conditions for the existence of an extension of orthogonally adjoint mappings to the lattice of ideals. Our results are illustrated by numerous examples.

We prove a refined quadratic normal approximation for the first absolute moment of normalized weighted Rademacher sums with bounded maximal coefficients. For any weight vector $w\in\mathbb{R}^{n}$ satisfying $\|w\|_{2}=1$ and $\|w\|_{\infty}\leq\beta$ with sufficiently small $\beta>0$, we establish the uniform error bound $|\mathbb{E}|\sum_{i=1}^{n}w_{i}X_{i}|-\sqrt{2/\pi}|=O(\beta^{2})$ over all admissible weight configurations. Our proof combines zero-bias Stein's method and refined small-ball probability estimates to exploit symmetry cancellation and control the non-smooth residual of the absolute-value test function. An explicit extremal construction further verifies the optimality of this quadratic convergence rate. As an application, we establish an asymptotically sharp refinement of the Friedgut–Kalai–Naor (FKN) theorem for Boolean functions, also known as the sharp $\pi/2$ theorem, characterizing the level-1 Fourier energy for functions deviating far from dictatorships.

In the spirit of Conway and Coxeter, we classify all $\mathbf{Y}$-frieze patterns of type $A_n$. In particular, we settle a conjecture made by de Saint Germain that all such $\mathbf{Y}$-frieze patterns arise from Conway-Coxeter frieze patterns. Moreover, our approach naturally leads to the enumeration of these $\mathbf{Y}$-frieze patterns in terms of Fuss-Catalan numbers.

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

The average performance of FirstFit online coloring on trees in the random order model is completely determined in recent works of Frei et al. and Bosek et al., showing $\Theta(\log n /\log\log n)$ number of colors, improving the $\Theta(\log n)$ colors in the adversarial model. We provide a few further results on slightly more general graph classes. Firstly, we extend their method to obtain a simple path-counting principle for sparse graph classes, which immediately yields for example that cactus graphs and uniform hypertrees exhibit a similar improvement. We then show that FirstFit uses only $O(1)$ colors on crown graphs, a standard example where adversarial arrival forces $\Theta(n)$ colors. We further show that density alone (even linear minimum degree) is insufficient to guarantee $O(1)$ colors even on bipartite graphs. Finally, we identify graph classes, including unit interval graphs and some graphs of high chromatic number, for which random arrival provides only limited improvement. We end with some open problems.

This document reports over 15 counterexamples found using GPT Pro over the course of 12 months, for problems in combinatorics, number theory, convexity, analysis, and other areas. The counterexamples concern both (reasonably) well-known problems as well as lesser known instances. The counterexamples are accompanied by the repo https://github.com/suvrit/count-ex-machina, and we warmly welcome new contributions toward helping building a rich “open-source countex library.”

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The anti-Ramsey number, denoted by $\ar(n,F),$ for a fixed graph $F$ and a positive integer $n$, is the maximum number of colors used in an edge-coloring of the complete graph $K_n$ that contains no rainbow copy of $F$. Meanwhile, the Turán number, denoted by $\ex(n,F),$ for graph $F$ and $n$, is the maximum number of edges in an $n$-vertex graph that does not contain $F$ as a subgraph. For a vertex $v$ and a multiset $\mathcal{H}$ of graphs, the suspension $\mathcal{H} + v$ of $\mathcal{H}$ is the graph obtained by connecting the vertex $v$ to all vertices of $H$ for each $H \in \mathcal{H}$. Let integers $k\ge 1$ and $r\ge 2$ be fixed, and suppose that $\mathcal{H}_{k+1}=\{H_1, H_2, \ldots, H_{k+1}\}+v$ satisfying $H_1, H_2, \ldots, H_{k+1}$ are pairwise vertex-disjoint edge-critical graphs, and $\chi(H_i)=r$ for $i=1,2,\ldots, k+1$.In this paper, we determine $ \ar(n,\mathcal{H}_{k+1}) $ for $k\ge 1$, $r\ge 2$ and sufficiently large $n$. This result unifies and generalizes a result of Liu et al. (arXiv:2411.08475) concerning the friendship graph, and a result of Lu et al. (arXiv:2507.13165) on the intersecting cliques.

For a finite graph, its energy is the sum of the absolute values of its adjacency eigenvalues. Let p and q be distinct odd primes with q at least 5, and let r be at least 1 and s be nonnegative. We determine the maximum energy among all integral circulant graphs whose order is p to the power 2r times q to the power 2s plus 1. The unique energy-maximising divisor set is the checkerboard set consisting of all divisors p raised to the power i times q raised to the power j, where i ranges from zero to 2r, j ranges from zero to 2s plus 1, and i plus j is even, and we obtain an explicit closed formula for the corresponding maximum energy. In particular, for q at least 5, when r and s are both equal to 1, our theorem establishes the conjectured maximality of Roldan's checkerboard divisor set and recovers his closed-form energy formula. The main ingredient is a semidefinite parity theorem for weighted prime-power Ramanujan transforms, proved by a parity-independent congruence reduction and a block Schur recurrence. Centring the divisor matrix then yields a sharp sign-matrix inequality from semidefinite bounds for a Kronecker-product operator. Analysing equality identifies the checkerboard pattern and proves uniqueness.

We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0<p<1$ and $\mathcal F\subseteq2^{[n]}$, define the $p$-biased measure by $\mu_p(\mathcal F)=\sum_{F\in\mathcal F}p^{|F|}(1-p)^{n-|F|}$. Two families $\mathcal F_1,\mathcal F_2\subseteq2^{[n]}$ are cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for every $F_1\in\mathcal F_1$ and $F_2\in\mathcal F_2$. We prove that, for every $n\geq t\geq 2$ and $0<p_1,p_2\leq1/(t+1)$, such a pair satisfies $\mu_{p_1}(\mathcal F_1)\mu_{p_2}(\mathcal F_2)\leq(p_1p_2)^t$. When $p_1,p_2<1/(t+1)$, equality holds if and only if both families are the same $t$-star $\mathcal S_T=\{F\subseteq[n]:T\subseteq F\}$ for some $T\in\binom{[n]}{t}$. Together with the previously known case $t=1$, this completely resolves Tokushige's measure conjecture. We also prove that every pair whose measure product is close to the maximum must be close to a common $t$-star. More precisely, if $p_1,p_2<1/(t+1)$ and $\mu_{p_1}(\mathcal F_1)\mu_{p_2}(\mathcal F_2)>(1-\varepsilon)^2(p_1p_2)^t$, then there exists $T\in\binom{[n]}{t}$ such that $\mu_{p_i}(\mathcal F_i\mathbin{\triangle}\mathcal S_T)<C\varepsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige's conjectured $C\sqrt{\varepsilon}$ estimate to $C\varepsilon$. For integer sequences, we prove that if every sequence in $\mathcal H_1\subseteq[m]^n$ agrees with every sequence in $\mathcal H_2\subseteq[m]^n$ in at least $t$ coordinates, then $|\mathcal H_1||\mathcal H_2|\leq m^{2(n-t)}$ for all $n\geq t\geq1$ and $m\geq t+1$. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection–union product theorem.

We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.

For a graph $G$, a proper coloring of $G$ is called proper conflict-free if for every non-isolated vertex $u$, there is at least one color appearing exactly once in $N_G(u)$. A graph $G$ is proper conflict-free $f$-choosable if for every list assignment $L$ with $|L(v)|\ge f(v)$ for each vertex $v$, $G$ admits a proper conflict-free $L$-coloring. Recently, Kashima, Škrekovski, and Xu proposed a conjecture on proper conflict-free list coloring. For a graph $G$, let $\kappa_G:V(G)\to \mathbb{N}$ be defined by \[ \kappa_G(v)= \begin{cases} 4, & \text{if } d_G(v)=2,\\[4pt] d_G(v)+1, & \text{if } d_G(v)\neq 2. \end{cases} \] They conjectured that every connected graph other than $C_5$ is proper conflict-free $\kappa_G$-choosable. In this paper, we confirm this conjecture in two classes of graphs with bounded average degree, thereby generalizing results of Kashima, Škrekovski, and Xu and of Wang and Zhang. We prove that every connected graph $G\neq C_5$ with either $\operatorname{mad}(G)<\frac{12}{5}$ or $\Delta(G)\le3$ is proper conflict-free $\kappa_G$-choosable. To prove these results, we introduce a method based on systems of proper conflict-free representatives and develop a construction of auxiliary graphs that preserves the maximum average degree bound.

We develop a method for evaluating finite and infinite series involving Catalan numbers by specializing alternating binomial identities at half-integer parameters. The key ingredient is an explicit representation of the generalized binomial coefficients $\binom{m+\frac12}{k+1}$ in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Bat{\i}r and Sofo, we derive several families of closed-form summation formulas involving Catalan numbers, harmonic numbers, and odd harmonic numbers. As special cases, we obtain evaluations of finite sums together with their corresponding infinite-series analogues. The approach provides a systematic mechanism for transforming identities for generalized binomial coefficients into identities for Catalan-number series.

Let $\mathcal F=\{2K_2,C_4,C_5\}$, so that $\operatorname{Forb}(\mathcal F)$ is the class of split graphs. Resolving a conjecture of Wang and Tang, we prove that the class of split graphs is generalized spectrally closed and that it admits no walk-realizable $\mathcal F$-supporter of any finite order.

A de Bruijn polyomino has colored cells and includes exactly one instance of each possible coloring of another polyomino with those colors. This generalizes the notion of a de Bruijn sequence. We are interested in the de Bruijn polyominoes of minimum size. As a step toward finding these, we introduce the problem of finding the smallest polyominoes containing at least $N$ translated copies of the polyomino $p$, allowing overlaps. We say these are $(p,N)$-dense and have size $a_{p,N}$. For certain pairs of polyominoes $p$ and $q$ of equal size, we show there are polyomino transformations relating $a_{p,N}$ and $a_{q,N}$. Using these transformations, we identify classes of polyominoes which share the sequence $(a_{p,N})_{N=1}^\infty$. We give closed forms for $(a_{p,N})_{N=1}^\infty$ for most polyominoes with up to five cells. Leveraging these results we introduce novel de Bruijn polyominoes, as well as other colored polyforms with de Bruijn-like properties.

Brenti conjectured that the Eulerian triangle is totally positive. In this paper, we prove that the Eulerian triangle is totally positive of order 3, thereby providing a partial affirmative answer to Brenti's conjecture. In addition, we establish total positivity of order 3 for several families of generalized Eulerian triangles.

We give solutions to several problems in combinatorial group theory recorded as open in the literature. Our main source is the Kourovka Notebook, although we also consider questions from elsewhere. All solutions were found with the assistance of large language models and checked by the authors.

We study integer-interval deformations of Coxeter arrangements of types $A$, $B$, and $D$, assigning an integer interval to each root direction through an integer-valued root function. We introduce a local compatibility condition on this root function. For compatible root functions, subject to additional local bounds in types $B$ and $D$, we derive characteristic-polynomial formulas as binomial sums over cyclic orders in type $A$ and signed permutations in types $B$ and $D$. The proofs use the finite-field method and reduce hyperplane avoidance to consecutive-gap inequalities. These formulas recover, in particular, the classical Catalan and Shi characteristic polynomials.