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$\beta$-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than…
We explore models where single new exotic states interact with the Standard Model through an asymmetric Standard Model portal with couplings to at least one gluon and one lepton. We consider the complete set of effective operators up to…
Many natural combinatorial quantities can be expressed by counting the number of homomorphisms to a fixed relational structure. For example, the number of 3-colorings of an undirected graph $G$ is equal to the number of homomorphisms from…
The cyclic sieving phenomenon provides a link between a polynomial analogue of Gauss congruence known as $q$-Gauss congruence, and a combinatorial analogue of Gauss congruence based on sequences of cyclic group actions. We strengthen this…
Two new classes of finite automata, called General hexagonal Boustrophedon finite automata and General hexagonal returning finite automata operating on hexagonal grids, are introduced and analyzed. The work establishes the theoretical…
A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively splits the set of classes into two subsets, and trains a binary classifier to distinguish…
Chordal clutters in the sense of [14] and [3] are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear…
We study the following variant of the 15 puzzle. Given a graph and two token placements on the vertices, we want to find a walk of the minimum length (if any exists) such that the sequence of token swappings along the walk obtains one of…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise…
Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…
We are often interested in decomposing complex, structured data into simple components that explain the data. The linear version of this problem is well-studied as dictionary learning and factor analysis. In this work, we propose a…
Each (equigenerated) squarefree monomial ideal in the polynomial ring $S=\mathbb{K}[x_1, \ldots, x_n]$ represents a family of subsets of $[n]$, called a (uniform) clutter. In this paper, we introduce a class of uniform clutters, called…
Analysis of the dynamics of the Dyck words helped solve the problem of representing the Catalan number as a sum of squares of natural numbers. In this case, the Dyck triangle is considered in different coordinates. In the calculations, we…
In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete…
New sets (typically found by computer search) with Sidon constant equal to the square root of their cardinalities are given. For each integer $N$ there are only a finite number of groups of prime order containing $N$-element extreme sets.…
We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial…
Statistical shape analysis of slabular objects like groups of hippocampi is highly useful for medical researchers as it can be useful for diagnoses and understanding diseases. This work proposes a novel object representation based on…
Two neural-network-based numerical schemes are proposed to solve the classical obstacle problems. The schemes are based on the universal approximation property of neural networks, and the cost functions are taken as the energy minimization…
A geometrical pattern is a set of points with all pairwise distances (or, more generally, relative distances) specified. Finding matches to such patterns has applications to spatial data in seismic, astronomical, and transportation…