2 research outputs found
Multi-latin squares
A multi-latin square of order and index is an array of
multisets, each of cardinality , such that each symbol from a fixed set of
size occurs times in each row and times in each column. A
multi-latin square of index is also referred to as a -latin square. A
-latin square is equivalent to a latin square, so a multi-latin square can
be thought of as a generalization of a latin square.
In this note we show that any partially filled-in -latin square of order
embeds in a -latin square of order , for each , thus
generalizing Evans' Theorem. Exploiting this result, we show that there exist
non-separable -latin squares of order for each . We also show
that for each , there exists some finite value such that for
all , every -latin square of order is separable.
We discuss the connection between -latin squares and related combinatorial
objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares
and -latin trades. We also enumerate and classify -latin squares of small
orders.Comment: Final version as sent to journa