Related papers: The four-way intersection problem for latin square…

Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…

We investigate the existence of closed polylines (also known as closed polygonal chains or self-crossing polygons) that intersect each of their edges the same number of times. The most general question in this corner of combinatorial…

Two Latin squares $L=[l(i,j)]$ and $M=[m(i,j)]$, of even order $n$ with entries $\{0,1,2,\ldots,n-1\}$, are said to be nearly orthogonal if the superimposition of $L$ on $M$ yields an $n\times n$ array $A=[(l(i,j),m(i,j))]$ in which each…

In this paper we study pattern avoidance in Latin Squares, which gives us a two dimensional analogue of the well studied notion of pattern avoidance in permutations. Our main results include enumerating and characterizing the Latin Squares…

A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the…

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k…

Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $\Theta$, the cardinality of the set $\mathcal{PLS}_{\Theta}$ of partial Latin squares which are…

Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of…

We answer a question posed by D\'enes and Keedwell that is equivalent to the following. For each order $n$ what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order $n$? We also…

The full $n$-Latin square is the $n\times n$ array with symbols $1,2,\dots ,n$ in each cell. In this paper we show, as part of a more general result, that any defining set for the full $n$-Latin square has size $n^3(1-o(1))$. The full…

Let $K$ be a field of char $K\neq 2$. For $a\in K$, we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial $X^4-aX^3-6X^2+aX+1$ over $K$ as the special case of the field intersection problem via…

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined…

Answering a recent question of Patchell and Spiro, we show that when a $d$-dimensional cube of side length $n$ is filled with letters, the word $\mathsf{CAT}$ can appear contiguously at most $(3^{d-1}/2)n^d$ times (allowing diagonals); we…

Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every…

This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.

Latin squares with a balance property among adjacent pairs of symbols---being "Roman" or "row-complete"---have long been used as uniform crossover designs with the number of treatments, periods and subjects all equal. This has been…

We prove that certain numbers occurring in a problem of paths enumeration, studied by Niederhausen (Catlan Traffic at the Beach, The Electronic Journal of Combinatorics, 9, (2002), 1--17), are top intersection numbers in the cohomology ring…

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…

In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem…

A partition on $[n]$ has a crossing if there exists $i\_1<i\_2<j\_1<j\_2$ such that $i\_1$ and $j\_1$ are in the same block, $i\_2$ and $j\_2$ are in the same block, but $i\_1$ and $i\_2$ are not in the same block. Recently, Chen et al.…