Overcolored Partition Restricted by Parity of the Parts
This paper extends the recently defined function , which counts multicolored partitions with specific color restrictions for even and odd parts, to the context of overpartitions.
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math.CO
292 papers
Almost All Vectorial Functions Have Trivial Extended-Affine Stabilizers
This paper proves that asymptotically almost all vectorial functions over finite fields possess trivial extended-affine stabilizers, implying that the number of equivalence classes matches the naive estimate and that random sampling is a highly effective strategy for cryptographic primitive design due to the exponential rarity of functions with nontrivial stabilizers.
Erd\H{o}s Matching (Conjecture) Theorem
This paper proves the long-standing Erdős Matching Conjecture, establishing that the maximum size of a family of -sized subsets of containing no pairwise disjoint sets is bounded by the larger of two canonical extremal constructions.
Classifying integer tilings and hypertilings
This paper classifies all tame integer tilings and hypertilings by developing a geometric model based on generalised Farey graphs in the hyperbolic plane, which connects these structures to triangulated polygons, friezes, and the Cayley hyperdeterminant.
Transversal and Hamiltonicity in a bipartite graph collection
This paper establishes improved minimum degree conditions that guarantee the existence of Hamiltonian transversals and Hamiltonian connectivity in both balanced and nearly balanced bipartite graph collections, thereby advancing previous results in the field.
Grid designs
This paper investigates the existence of -designs (decompositions of complete graphs into edge-disjoint copies of a grid graph ), proving that such designs exist for toroidal grids when is an odd prime or its square, and for the path-grid (which relates to scrambling Connections puzzles), while showing that admits no such design.
Simplex volumes in hyperplane arrangements
This paper investigates dual variants of the Erdős distinct distances and unit distance problems by analyzing the maximum numbers of unit, extremal, and distinct-volume -simplices formed by arrangements of hyperplanes in .
Dimension statistics of representations of finite groups
This paper investigates the dimension statistics of representations for finite groups, demonstrating that for reductive groups over finite fields and symmetric groups, quantities such as representation dimensions and conjugacy class sizes exhibit asymptotically constant or log-constant behavior as the field size or group rank tends to infinity.
Three formulas for CSM classes of open quiver loci
This paper presents one geometric and two combinatorial formulas for computing the equivariant Chern-Schwartz-MacPherson classes of open quiver loci in type quiver representations, introducing "chained generic pipe dreams" and providing streamlined versions of known formulas for the associated quiver polynomials.
On the Diameter of Arrangements of Topological Disks
This paper establishes that the diameter of the dual graph of an arrangement of topological disks is bounded by a function of and the maximum number of connected components in any pairwise intersection, providing a tight bound of for two disks and an bound for disks by analyzing the count of maximal faces.
There is no prime functional digraph: Seifert's proof revisited
This paper provides an accessible, modernized proof of Ralph Seifert's 1971 result that no prime functional digraphs exist, thereby resolving a recent conjecture by Antonio E. Porreca.
K-promotion on m-packed labelings of posets
This paper extends Pechenik's K-theoretic promotion operator () from tableaux to general posets and rooted trees, establishing divisibility properties for orbit sizes and completely determining these sizes for specific tree structures under certain conditions.
Ordinarization numbers of numerical semigroups
This paper investigates the enumeration of numerical semigroups of genus with a fixed ordinarization number by interpreting the problem as counting integer points in rational polyhedral cones using Ehrhart theory, while deriving specific formulas and geometric characterizations for cases involving ordinarization numbers 1 and 2, two-generated semigroups, supersymmetric semigroups, and interval-generated semigroups.
Graph quandles: Generalized Cayley graphs of racks and right quasigroups
This paper establishes a geometric group theory framework for right quasigroups by introducing graph markings and invariants to characterize their Cayley (di)graphs, thereby proving that all racks are realizable by their full Cayley graphs and providing graph-theoretic characterizations for various algebraic structures.
Infinite linear patterns in sets of positive density
This paper characterizes all possible infinite linear configurations that exist within the shift of any set of positive upper Banach density, thereby unifying and generalizing Szemerédi's theorem on arithmetic progressions and the recent density finite sums theorem.
Rigidity of polytopes with edge length and coplanarity constraints
This paper investigates a novel notion of polytope rigidity where edge lengths and face planarity are preserved while face shapes may vary, proving that convex polytopes are generically rigid in dimension three and conjecturing this holds for all dimensions .
A Path Variant of the Explorer Director Game on Graphs
This paper investigates a path-based variant of the Explorer-Director game on graphs, demonstrating that the resulting number of visited vertices can differ arbitrarily from the original distance-based version depending on the graph structure.
Representability of the direct sum of uniform q-matroids
This paper demonstrates that while the direct sum of general representable -matroids is not necessarily representable, the direct sum of uniform -matroids is always representable over a sufficiently large field, a result established through algebraic and geometric tools involving cyclic flats.
Some polynomial classes for the acyclic orientation with parity constraint problem
This paper identifies three necessary conditions for the existence of acyclic T-odd orientations, defines and characterizes polynomial graph classes where these conditions are sufficient, and provides constructive polynomial-time algorithms to build such orientations for these classes and their Cartesian products.
The Hofstadter consecutive-sum sequence omits infinitely many positive integers
This paper resolves a conjecture from OEIS entry A005243 by proving that the greedy self-generating sequence defined by Hofstadter, where each term is the smallest integer greater than the previous one representable as a sum of at least two consecutive earlier terms, omits infinitely many positive integers.