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Indivisible partitions of latin squares

Indivisible partitions of latin squares,10.1016/j.jspi.2010.06.020,Journal of Statistical Planning and Inference,Judith Egan,Ian M. Wanless

Indivisible partitions of latin squares   (Citations: 4)
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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 0ck. For orders n∉{2,6}, 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 1kn then there exists a latin square of order n with a partition into indivisible k-plexes.Define κ(n) 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 κ(n) exactly for n≤8 and find that κ(9)∈{6,7}. Up to order 8 we count all indivisible partitions in each species.For each group table of order n≤8 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 ⌊n/2⌋ disjoint 2-plexes.By extending an argument used by Mann, we show that for all n≥5 there is some k∈{1,2,3,4} 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.
Journal: Journal of Statistical Planning and Inference - J STATIST PLAN INFER , vol. 141, no. 1, pp. 402-417, 2011
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    • ...Such conjectures and the evidence from small order examples [5] suggest that all latin squares are ‘highly divisible’ in the sense that they possess many different partitions...
    • ...It would also be particularly interesting to know the asymptotic behaviour of κ(n) .I n [5] it is found that κ(n )> 1 n at least for 5 ≤ n ≤ 9, so there may be room to improve...
    • ...An extension to Theorem 1.2 is shown in [5]...

    Darryn E. Bryantet al. Indivisible plexes in latin squares

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