Indivisible partitions of latin squares

In a latin square of order n, a k-plex is a selection of kn entries in which each row, column and symbol occurs k times. A 1-plex is also called a transversal. An indivisible k-plex is one that contains no c-plex for 0ocok. For orders n= 2f2,6g, existence of latin squares with a partition into 1-plexes was famously shown in 1960 by Bose, Shrikhande and Parker. A main result of this paper is that, if k divides n and 1okon then there exists a latin square of order n with a partition into indivisible k-plexes. Define kðnÞ to be the largest integer k such that some latin square of order n contains an indivisible k-plex. We report on extensive computations of indivisible plexes and partitions in latin squares of order at most 9. We determine kðnÞ exactly for nr8 and find that kð9Þ 2 f6,7g. Up to order 8 we count all indivisible partitions in each species. For each group table of order nr8 we report the number of indivisible plexes and indivisible partitions. For group tables of order 9 we give the number of indivisible plexes and identify which types of indivisible partitions occur. We will also report on computations which show that the latin squares of order 9 satisfy a conjecture that every latin square of order n has a set of bn=2c disjoint 2-plexes. By extending an argument used by Mann, we show that for all nZ5 there is some k 2 f1,2,3,4g for which there exists a latin square of order n that has k disjoint transversals and a disjoint (n k)-plex that contains no c-plex for any odd c. & 2010 Elsevier B.V. All rights reserved.