Indivisible partitions of latin squares Judith Egan Ã , Ian M. Wanless School of Mathematical Sciences, Monash University, Victoria 3800, Australia article info Article history: Received 30 October 2009 Received in revised form 16 June 2010 Accepted 16 June 2010 Available online 22 June 2010 Keywords: Latin square Transversal Plex Indivisible partition abstract 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 0 oc ok. 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 1 ok on 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 n r8 and ﬁnd that kð9Þ2f6, 7g. Up to order 8 we count all indivisible partitions in each species. For each group table of order n r8 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 n Z5 there is some k 2f1, 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. 1. Introduction A latin square of order n is an n  n matrix in which n distinct symbols are arranged so that each symbol occurs once in each row and column. A latin square of order n can be speciﬁed by a set L of n 2 ordered triples ðx, y, zÞ2 I ðLÞ 3 , where I ðLÞ is a set of cardinality n, and no two distinct elements of L agree in more than one coordinate. We say that L is indexed by I ðLÞ. If (x,y,z) is in L, then the corresponding matrix has the symbol z at the intersection of row x and column y. In a latin square L of order n,a k-plex is a subset of L consisting of kn entries in which each row, column and symbol occurs k times. We call a k-plex an odd plex or an even plex if k is odd or even, respectively. Various names for speciﬁc cases of k-plexes have been used. A transversal of a latin square corresponds to k = 1. Statistical literature sometimes refers to a transversal as a directrix and uses the terms duplex, triplex and quadruplex for a 2-plex, 3-plex and 4-plex respectively. Other names are mentioned by Wanless (2002). Following Bryant et al. (2009) and Egan and Wanless (2008), we differ from Wanless (2002) in our deﬁnition of a k-plex as we insist on containment in some latin square of order n. We call a set of plexes parallel, or disjoint, if no two of them share a common element. The union of an a-plex and a parallel b-plex in a latin square L yields an (a + b)-plex of L. The reverse process, that is dividing an (a + b)-plex into an a-plex Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jspi Journal of Statistical Planning and Inference 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.06.020 Ã Corresponding author. E-mail addresses: judith.egan@monash.edu, judith.egan@gmail.com (J. Egan), ian.wanless@monash.edu (I.M. Wanless). Journal of Statistical Planning and Inference 141 (2011) 402–417