Papers

Permutation pattern avoidance on Ferrers boards has become a central topic in enumerative combinatorics with important connections to matchings, set partitions, and other combinatorial structures as it allows one to build families of Wilf-equivalent patterns. While shape-Wilf-equivalence classes have been completely determined for individual patterns and pairs of patterns of length three, the corresponding classification for larger pattern sets has remained open. In this paper, we provide a complete classification of the shape-Wilf-equivalence classes of triples of patterns of length three. Our proofs use a bijective encoding of pattern avoiding transversals to establish all equivalence classes. As an application, we enumerate matchings avoiding triples of patterns of length three for all but two equivalence classes, extending previous results of Bloom and Elizalde. These enumerative results identify additional families of combinatorial objects counted by the Fuss-Catalan numbers and by other integer sequences appearing in the OEIS.

This paper establishes a formal framework for the cocyclic development of Hadamard matrices over loops based on a novel cohomology theory with associativity obstructions. In this context, we formalize the notion of a loop-cocyclic Hadamard matrix, so that both the classical cocyclic Hadamard matrices over groups and the recently introduced pseudococyclic Hadamard matrices over loops are naturally embedded within this broader algebraic architecture. To validate the computational viability of this framework, we prove the existence of four new Hadamard equivalence classes of order 24 and eight new classes of order 28 that are strictly non-cocyclic over any finite group, arising uniquely as loop-cocyclic developments over Moufang and right Bol loops.

Let $G$ be a simple graph with maximum degree $\Delta\ge 3$, and let $P(G,k)$ denote its chromatic polynomial. For each positive integer $k$, the list-color function $P_{\ell}(G,k)$ is the minimum number of $L$-colorings of $G$ over all $k$-assignments $L$. In this paper, we prove that $P_{\ell}(G,k)=P(G,k)$ for every integer $k\ge 23.41\Delta$. This gives a threshold for equality that is linear in the maximum degree and independent of the number of vertices or edges. It improves the known sufficient condition $k\ge |E(G)|-1$ for graphs with sufficiently many edges relative to their maximum degree.

For $n\geq1$, choose one residue class $r_k\pmod{k}$ for every $2\leq k\leq n$, and retain those $m\in\{2,\ldots,n+1\}$ for which $m\not\equiv r_k\pmod{k}$ whenever $k<m$. Let $N(n)$ be the number of distinct survivor sets obtained in this way. We prove \[ \log 2\leq \liminf_{n\to\infty}\frac{\log N(n)}{n/\log n} \leq \limsup_{n\to\infty}\frac{\log N(n)}{n/\log n} \leq2\log 2. \] If $A_{\rm adm}(n)$ denotes the number of subsets of $\{2,\ldots,n+1\}$ that omit at least one residue class modulo every prime, then our main comparison is \[ \log N(n)=\log A_{\rm adm}(n)+o\!\left(\frac{n}{\log n}\right). \] We also obtain an asymptotic formula for the number of distinct intersections of survivor sets with the primes, and an exact recurrence $N(n+1)=N(n)+E(n)$, where $E(n)$ counts the survivor sets that can be extended by the new point $n+2$.

This work provides a complete analytical characterization of the transfer- matrix structure associated with the finite Kronig-Penney model consisting of one-dimensional arrays of Dirac delta potentials recently introduced by Figueroa et al. (2025). Although their study identified the emergence of specific transfer-matrix entries and related combinatorial coefficients, a rig- orous derivation of the corresponding closed-form expressions has not yet been established. By expressing the N th power of the unit-cell transfer ma- trix in terms of Chebyshev polynomials of the second kind, we obtain explicit closed-form representations for the global transmission and reflection ampli- tudes. The proposed formulation reveals a previously unrecognized structural correspondence between multiple quantum scattering processes, discrete con- volutional patterns, and hypercomplex combinatorial structures.

Exceptional CR-quadrics are studied. An example of an exceptional (4,4)-quadric is constructed; it realizes the minimum exceptional type with respect to both n and k. Its graded Lie algebra is described. The available information on exceptional types is summarized, and the lattice of CR-types is decomposed into the disjoint union of three sets: A, the types for which exceptional quadrics are impossible; B, the types for which examples of exceptional quadrics are known; and C, the types whose status is currently unknown (neither an example nor a nonexistence proof is known). Several questions are posed.

We introduce and begin the study of sequence b-colorings, a natural generalization of the classical notion of b-colorings introduced by Irving and Manlove in 1999. In a sequence b-coloring, each color class is required to contain a prescribed minimum number of color-dominating vertices (CDVs). We establish several fundamental properties of the associated parameters, prove that every sequence is realizable, and show that the problem of deciding whether a particular graph realizes a particular sequence is NP-complete. We also characterize the sequences realized by cycles, obtain results on regular graphs with prescribed girth, and investigate colorings requiring one additional CDV, including a characterization of connected graphs with chromatic number $3$ for which no such coloring exists.

Let $(W,S)$ be a Coxeter system and let $J\subseteq S$. The right $J$-ascent set $$ A_R^J(w)=\{s\in S\mid ws\in{}^JW,\ \ell(ws)=\ell(w)+1\} $$ is a natural refinement of the classical ascent set adapted to the parabolic quotient ${}^JW$. We establish a local transition formula describing the behaviour of $J$-ascent sets under right multiplication by a simple reflection. As a consequence, for every $w\in{}^JW$ and every $s\in A_R^J(w)$, we prove that $$ \Bigl||A_R^J(w)|-|A_R^J(ws)|\Bigr| \leq \max\{1,\text{deg}(s)-1\}. $$

Consider a random geometric graph with vertices given by a Poisson point process, and whose edges depend on independent marks corresponding to the vertices and pairs of vertices. In this paper, we study two related questions on this general model: the number of edge crossings in a projection of this graph, and its graph-theoretical planarity. We focus on models with heavy-tailed mark distributions, in particular with polynomial tails with arbitrary exponents. We show that the asymptotic behaviour varies significantly depending on this exponent.

Tutte's $3$-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In 2013, Lovász, Thomassen, Wu, and Zhang proved that every odd-$7$-edge-connected graph admits a nowhere-zero $3$-flow; consequently, the conjecture holds for $4$-edge-connected graphs with no edge-cut of size $5$. We consider the complementary situation in which $5$-edge-cuts are allowed. We show that, when edge-cuts of size $5$ are permitted, Tutte's $3$-flow conjecture holds for graphs with no edge-cut of any size from $6$ to $k$, where $k$ is an absolute constant. In fact, $k=40$ suffices and we prove a stronger version in which only nontrivial edge-cuts of those sizes are forbidden. A graph is called essentially $t$-edge-connected if deleting any set of at most $t-1$ edges leaves at most one nontrivial component. Motivated by Jaeger's weak $3$-flow conjecture, we prove an analogous result for essential edge connectivity: every $4$-edge-connected, essentially $41$-edge-connected graph admits a nowhere-zero $3$-flow.

Recently, Kang, Kim, and Lee \cite{Kang2026} developed a unified moment-trace framework for symmetric partition statistics using complete Bell polynomials and their inversion formula. In this paper, we apply this framework to crank statistics for $t$-core partitions and overpartitions. For $t\in\{5,7,11,17,19\}$, we show that the normalized even crank moment generating functions for $t$-core partitions admit partition-trace representations in terms of the functions $D^{(t)}_{2s}(\tau)$, together with suitable Bernoulli-number shifts. We also establish inverse trace formulas that recover $D^{(t)}_{2s}(\tau)$ from the corresponding normalized even crank moments. For overpartitions, we obtain analogous trace and inverse-trace identities for the normalized even moments associated with the first and second residual crank generating functions. As applications, we use complete Bell polynomials and their inversion formula to obtain explicit expressions for the $t$-core partition numbers and overpartitions number in terms of sums involving divisor function.

We investigate the properties of matrices obtained from a geometric transformation of the first $N$ rows of Pascal's triangle. For $N > 2$, their congruence properties form a Pascal tiling, that is, a perfect alternation between entries congruent to $0 \pmod{N}$ and the others, if and only if $N$ is prime. This result yields an alternative proof of the classical congruence $L_N-1\equiv 0 \pmod{N}$ for prime $N$, where $L_N$ denotes the $N$th Lucas number. Within the framework of the Pascal tiling theorem, this congruence can be expressed as a sum of entries lying along a diagonal of one of the matrices under consideration; when $N$ is prime, each of these entries is congruent to $0 \pmod{N}$. By contrast, for Fibonacci pseudoprimes, the sum remains congruent to $0 \pmod{N}$ while at least one of its terms is not. Finally, these results are interpreted in terms of decompositions of binomial coefficients and extended to multinomial coefficients, leading to a study of the associated symmetries. This perspective highlights the case where $N$ is a prime power and clarifies the conditions under which a Pascal tiling arises.

Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erd\H{o}s. In the 1980s, Erd\H{o}s and S\'os asked to determine Tur\'an densities of $K_4^{(3)-}$ (broken tetrahedron) and $K_4^{(3)}$ (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Tur\'an density of $K_4^{(3)}$ is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.

We prove that no strongly regular graph with parameters $(266, 45, 0, 9)$ exists. The proof is formalized in Lean 4 and Mathlib without external infeasibility certificates or assumed classification theorems. A hypothetical graph gives a rank-$12$ integral Gram lattice with an integral centroid. A Lorentzian change of form, a marked $D_7$ gluing, and an explicit rank-six complement produce a positive-definite even unimodular lattice of rank $24$, together with the original indexed family of $220$ vectors. Harmonic theta identities and a root-isolation inequality force the root system $A_{11} \perp D_7 \perp E_6$. First and second moments then exclude the possible complements: the final case reduces to an impossible binary projection identity $4x + 4y - 2z = 50$. A type-$A$ subcase is closed by a separate classification-free proof of the known nonexistence of a quasi-symmetric $2$-$(56, 12, 9)$ design with intersections $0, 3$. That argument constructs a Krein graph and forces a Steiner $3$-$(12, 4, 1)$ design, contradicting its replication equation. The formal theorem depends only on the three standard Lean axioms and has also been checked independently with nanoda. The archived formalization is release v2.0.0.

We characterize the uniform theta graph $\Theta(t,2)$ by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph $D_n^{(t)}$ is $2n$-periodic and that the uniform theta graph $\Theta(t,n)$ is $(2n+2)$-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely $D_3^{(t)}$ with $t \geq 2$ and $\Theta(t,2)$ with $t \geq 1$.

We give a computer-assisted proof that every finite union-closed family containing a nonempty set has an element in at least 0.38288525 of its members. The argument combines independent sampling with conditionally independent sampling and an explicit product lower bound for a binary-entropy kernel. We identify the limiting constant of this pointwise method through a one-variable stationary equation. We also construct a global coupling by balancing inverse union multiplicities and prove a strictly positive, quantitative entropy gain over independent sampling. Combining the two arguments gives an additional frequency bound depending on the size of the family.

A balanced spanning tree is a spanning tree that contains an edge whose removal partitions the vertices into exactly two connected subtrees of equal size. In this paper, we establish explicit recurrence relations for the number of spanning trees in $2 \times n$ triangular strip lattices- obtained by adding a diagonal edge to each square of a $2 \times n$ grid graph- generalizing combinatorial counting techniques introduced by Raff [Raf08]. We then adapt arguments of Gallagher and Tapp [GT25] to count balanced spanning trees of arbitrary triangular strip lattices. We establish sharp asymptotic bounds for the proportion of balanced spanning trees as $n \rightarrow \infty$. Finally, we determine the probability that a spanning tree of a $2 \times n$ triangular strip lattice chosen uniformly at random is balanced as $n \rightarrow \infty$.

For every integer $r\ge4$, and $\rho \in [0,1]$, we asymptotically determine the maximum proportion of $r$-element sets of vertices that induce either a clique or an independent set in a large graph with density $\rho$. This generalizes a result of Olpp for $r=3$. After the initial idea for the main proof was found by the authors, various AI models were used to streamline the argument and perform the calculations necessary for completion of the proof.

For two distinct binary words of length $n$, the separating words problem asks for a small deterministic finite automaton that accepts exactly one of them. Chase proved a $\widetilde O(n^{1/3})$ upper bound using a complex-analytic estimate for sparse polynomials. We replace that estimate by a finite-difference argument and a second-order real recurrence cutoff. The resulting elementary proof gives an explicit bound of $O(n^{1/3}(\log n)^{7/3})$ states.

A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $\kappa(\alpha)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $\alpha$. For $k \ge 2$ we show: \begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \, \kappa\!\left(1 - \frac{1}{2r}\right) \, . \end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \begin{align} \left(\tfrac{1}{2}-o(1)\right) \, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \, \frac{\log_2 k}{k} \, . \end{align}

For an ordered graph $H$ and an integer $q \geq 2$, let $r_{<}(H;q)$ denote the $q$-color ordered Ramsey number of $H$. Conlon, Fox, Lee and Sudakov asked whether, for every $q \geq 3$, there is a constant $c_q$ such that $r_{<}(M;q) \leq n^{c_q\log n}$ for every ordered matching $M$ on $n$ vertices. We answer this negatively in a strong form: for every $q \geq 2$, there is $c_q>0$ such that almost every perfect matching $M$ on $[n]$ satisfies $r_{<}(M;q)>2^{c_q(\log n)^q/(\log\log n)^{q-1}}$. This matches the general upper bound up to a factor of $(\log\log n)^{q-1}$ in the exponent. We also give two applications. First, we prove the lower bound conjectured by Fox, He and Wigderson for multicolor Ramsey numbers of acyclic digraphs of bounded degree. Second, we strengthen a result of Axenovich, Rollin and Ueckerdt by giving a superquasipolynomial lower bound on the maximum chromatic number of $M$-free ordered graphs, for almost every perfect matching $M$ on $[n]$.

Let \(G\) be a graph with \(m\) edges and adjacency spectral radius\(\rho(G)\), and let \(N(C_{2k+1},G)\) denote the number of copies of \(C_{2k+1}\) in \(G\). For each fixed integer \(k\ge 2\), define \( g_k(m):=\frac{k-1+\sqrt{4m-k^2+1}}{2}. \) Li, Zhai and Shu [European J. Combin., 2024] determined the spectral extremal threshold for odd cycles by proving that, for all sufficiently large \(m\), every \(C_{2k+1}\)-free graph \(G\) with \(m\) edges satisfies \(\rho(G)\le g_k(m)\). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer \(k\ge 2\), we prove that \[ \inf_{\substack{e(G)=m\\ \rho(G)>g_k(m)}} \frac{N(C_{2k+1},G)}{m^k} = \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}+o(1) \qquad\text{as }m\to\infty. \] Thus every \(m\)-edge graph whose spectral radius exceeds the \(C_{2k+1}\)-free threshold contains at least \[ \Big( \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}-o(1) \Big)m^k \] copies of \(C_{2k+1}\), and the leading constant is asymptotically best possible. In particular, taking \(k=2\), we obtain if \( \rho(G)>\frac{1+\sqrt{4m-3}}{2}\) then \( N(C_5,G)\ge \left(\frac{2}{9}-o(1)\right)m^2, \) with the constant \(2/9\) being asymptotically optimal. This answers a question of Chen, Li and Tang concerning the existence and the largest possible value of a constant \(C>0\) for which the same spectral condition guarantees at least \(Cm^2\) copies of \(C_5\). More generally, our result resolves a recent problem of Li, Lin, Liu and Zhang on spectral supersaturation for odd cycles. The proof combines spectral stability and resolvent analysis with estimates for odd spectral moments and a careful treatment of non-injective closed walks.

We prove the conjecture of Borowiecki and Broere that every finite simple planar graph admits a generalized total coloring with four colors: a proper vertex coloring with four colors together with an edge coloring with four colors in which each edge color class is a forest and no edge receives the color of either endpoint. In fact every proper four-coloring of such a graph extends to an edge coloring of the required kind. The proof combines a component-count inequality for planar graphs whose vertices are partitioned into four independent sets, obtained by completing to a triangulation, with the matroid partition theorem applied to four graphic matroids. A Lean 4 formalization relative to six named background assumptions is described below.

We give a counterexample to Wormald's conjecture. A 16-vertex cubic graph, the disjoint union of $K_{3,3}$ with a 10-vertex bridged graph built from two subdivided copies of $K_4$, has no partition of its edges into two isomorphic spanning linear forests. The obstruction is parity: any such partition would force every monochromatic component to have even order while requiring a five-vertex side of a bridge to be partitioned into such components. Adjoining copies of $K_4$ gives counterexamples in every order $16+4t$. The counterexample is disconnected and leaves the connected case unresolved. The full family is verified in Lean without custom axioms.

Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, the exact value of $S^k_{2r}(n)$ was recently determined by Dumitrescu and the authors for all sufficiently large $n$ when $k=3$, and conditionally on an optimization problem when $k\geq4$. In this paper we establish a sharp estimate in odd dimensions by proving that \[S^k_{2r+1}(n)=\binom{r}{k}\left(\frac{n}{r}\right)^k+\Theta(n^{k-2/3})\] for all fixed $r\geq k\geq3$. This can be viewed as a generalization of the result by Erdős and Pach on unit distances in odd dimensions. We also establish a structural characterization of nearly extremal point sets. A notable difference from the even-dimensional case is that nearly extremal point sets in odd dimensions admit two distinct configurations. The proof leverages techniques from hypergraph Turán theory and linear algebra, with additional geometric arguments.

This book develops algebraic and combinatorial methods for studying discrete structures. It brings together graph theory, Boolean functions, Fourier analysis on finite groups, coding theory, perfect colorings and perfect codes, association schemes, Latin squares, and related topics. A central theme is the interaction between different representations of the same object: combinatorial, algebraic, spectral, and coding-theoretic. The main mathematical object studied in this book is a perfect coloring of a graph, or, equivalently, an equitable partition of a graph. The book is intended for advanced undergraduate and graduate students in mathematics and computer science, as well as for researchers in discrete mathematics, combinatorics, coding theory, and related fields.

We give a short, conceptual proof that prime graphs have at most two transitive orientations, a much-quoted result of Gallai. Our proof uses chains, introduced by Chudnovsky, Kim, Oum, and Seymour, which provide a transparent characterization of primality. Transitivity induces a forcing relation on edges; using chains, we show that any two edges of a prime graph are equivalent under this relation, and thus any transitive orientation is unique up to reversal.

Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n} \rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.

Let $r \ge 2$ and $t \ge 1$. It is shown that if $T$ is an $r$-uniform tight tree with $t$ edges and $H$ is a $T$-free $r$-uniform hypergraph, then $|E(H)|\le (t-1)|\sh H|/r$, where $\sh H$ is the $(r-1)$-shadow of $H$. \iffalse Equality holds only for $(n,t + r - 2,r)$-designs.\fi The bound is tight infinitely often, and establishes Kalai's Conjecture, whose $r=2$ case is the Erd\H os-Sós Conjecture. The proof was found by GPT-6 Astra, extending its method of proof for the Erd\H os-Sós conjecture to the hypergraph setting. It is noteworthy that previous proofs of special cases of the Erd\H os-Sós conjecture do not extend to give tights bounds in the hypergraph setting. A strengthening of the Erd\H os-Sós conjecture due to the authors about tight lower bounds on the number of copies of a tree in a graph with average degree $d \ge t-1\ge 0$ remains open.

We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of Mütze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by Barrus, in the strongest form either admits. The argument is an induction on the number of ground vertices, cutting the realization graph at a single ground vertex into fibers and the quotient they lie over. The proof is formalized in Lean 4 and checked by its kernel, with seven results cited from the literature and nothing else assumed. Its engine is a classification. The realizable neighborhoods of a ground vertex form a shifted family – one closed under replacing an element by a smaller one – and the quotient is the Johnson graph of that family. Such a Johnson graph can fail to be Hamilton-connected, and we determine exactly when: the failures are one explicit family of examples, the Y-families, and each of them fails between a single pair of its members. A shifted family with a greatest member never fails, and those families are exactly the shifted matroids, where the conclusion already follows from the theorem of Naddef and Pulleyblank on the graphs of 0/1-polytopes. The obstruction lives entirely outside the matroid case, which is why it has not been met before.