Related papers: Conditions to Extend Partial Latin Rectangles

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely…

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that…

Semi-Latin squares have been extensively studied. They can be interpreted as a special case of latinized block designs where the number of columns is equal to the number of replicates in the design. Latinized row-column designs are…

We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. It follows from these constructions that there is a group-based complete Latin square of order $n$…

An arrangement of s elements in s rows and s columns, such that no element repeats more than once in each row and each column is called a Latin square of order s. If two Latin squares of the same order superimposed one on the other and in…

Let $L(n)$ be the number of Latin squares of order $n$, and let $L^{\textrm{even}}(n)$ and $L^{\textrm{odd}}(n)$ be the number of even and odd such squares, so that $L(n) = L^{\textrm{even}}(n) + L^{\textrm{odd}}(n)$. The Alon-Tarsi…

A classical question in combinatorics is the following:\ given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of \textbf{$\epsilon$-dense partial Latin squares}:\ partial…

In 1782, Euler conjectured that no Latin square of order $n\equiv 2\; \textrm{mod}\; 4$ has a decomposition into transversals. While confirmed for $n=6$ by Tarry in 1900, Bose, Parker, and Shrikhande constructed counterexamples in 1960 for…

Partial difference sets (for short, PDSs) with parameters ($n^2$, $r(n-\epsilon)$, $\epsilon n+r^2-3\epsilon r$, $r^2-\epsilon r$) are called Latin square type (respectively negative Latin square type) PDSs if $\epsilon=1$ (respectively…

For Latin squares the units (rows and columns) have fixed sum. The same holds for rows, columns, and blocks in Sudokus. Summing the elements of a unit yields a linear equation, and the set of all such equations forms a system of linear…

A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $\epsilon>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper…

This paper deals with distinct computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this…

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an…

Latin squares are well studied combinatorial objects. In this paper we generalize the concept and propose new objects like Latin triangles, free Latin squares, Latin tetrahedra, free Latin cubes, etc. We start with a classic definition of…

We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We…

A Latin hypercuboid of order $n$ is a $d$-dimensional matrix of dimensions $n\times n\times\cdots\times n\times k$, with symbols from a set of cardinality $n$ such that each symbol occurs at most once in each axis-parallel line. If $k=n$…

Let $T(n)$ denote the maximal number of transversals in an order-$n$ Latin square. Improving on the bounds obtained by McKay et al., Taranenko recently proved that $T(n) \leq \left((1+o(1))\frac{n}{e^2}\right)^{n}$, and conjectured that…

This article, showing that almost all objects in the title are asymmetric, is re-typed from a manuscript I wrote somewhere around 1980 (after the papers of Bang and Friedland on the permanent conjecture but before those of Egorychev and…

We use a greedy probabilistic method to prove that for every $\epsilon > 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-\epsilon)n$. That is, we show the existence of a second Latin rectangle such…

We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we 1. Determine the number of orthogonal mates for each species of latin square of order $n$. 2. Calculate the…