A latin square of order n possessing a cyclic automorphism of order n is said to be diagonally cyclic because its entries occur in cyclic order down each broken diagonal. More generally, we consider squares possessing any cyclic automorphism. Such squares will be named after Parker, in recognition of his seminal contribution to the study of orthogonal latin squares. Our primary aim is to survey...

Extended Abstract. It is a traditional way in biological, sociological, agricultural and geological studies to partition a geographical area into quadrats and then take a sample of them by a particular sampling design. We study the relevant characteristic of quadrats to estimate a parameter of the population. We suppose that the variable of interest has a positive spatial autocorrelation. Sampl...

In this note we introduce the concept of the trade space of a latin square. Computations using Sage and the GAP package Simplicial Homology are presented.

It is shown that any partial Latin square of order at least six which consists of two filled rows and two filled columns can be completed.

A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that bn ≤ T (n) ≤ ...

We consider the Youden square formed by deleting one row of a v×v back-circulant Latin square and establish a critical set that contains a number of elements which is equal to v/4 (v even) or (v − 1)/4 (v odd). We show that this critical set is minimal, for v even.

This paper gives a general method for constructing quasi-Latin square, quasi-Latin 12 rectangle and extended quasi-Latin rectangle designs for symmetric factorial experiments. Two 13 further methods are given for parameter values satisfying certain conditions. Designs are 14 constructed for a range of numbers of rows and columns so that the different construction 15 techniques are illustrated. ...

In this paper we focus on the representation of Steiner trades of volume less than or equal to nine and identify those for which the associated partial latin square can be decomposed into six disjoint latin interchanges. 1 Background information In any combinatorial configuration it is possible to identify a subset which uniquely determines the structure of the configuration and in some cases i...

An m-flower in a latin square is a set of m entries which share either a common row, a common column, or a common symbol, but which are otherwise distinct. Two m-flowers are disjoint if they share no common row, column or entry. In this paper we give a solution of the intersection problem for disjoint m-flowers in latin squares; that is, we determine precisely for which triples (n,m, x) there e...