Title: Sesqui-arrays, a generalisation of triple arrays,
Abstract: A triple array is a rectangular array containing letters, each letter
occurring equally often with no repeats in rows or columns, such that the
number of letters common to two rows, two columns, or a row and a column are
(possibly different) non-zero constants. Deleting the condition on the letters
common to a row and a column gives a double array. We propose the term
\emph{sesqui-array} for such an array when only the condition on pairs of
columns is deleted. Thus all triple arrays are sesqui-arrays.
In this paper we give three constructions for sesqui-arrays. The first gives
$(n+1)\times n^2$ arrays on $n(n+1)$ letters for $n\geq 2$. (Such an array for
$n=2$ was found by Bagchi.) This construction uses Latin squares. The second
uses the \emph{Sylvester graph}, a subgraph of the Hoffman--Singleton graph, to
build a good block design for $36$ treatments in $42$ blocks of size~$6$, and
then uses this in a $7\times 36$ sesqui-array for $42$ letters.
We also give a construction for $K\times(K-1)(K-2)/2$ sesqui-arrays on
$K(K-1)/2$ letters. This construction uses biplanes. It starts with a block of
a biplane and produces an array which satisfies the requirements for a
sesqui-array except possibly that of having no repeated letters in a row or
column. We show that this condition holds if and only if the \emph{Hussain
chains} for the selected block contain no $4$-cycles. A sufficient condition
for the construction to give a triple array is that each Hussain chain is a
union of $3$-cycles; but this condition is not necessary, and we give a few
further examples.
We also discuss the question of which of these arrays provide good designs
for experiments.