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dc.contributor.authorBailey, Rosemary Anne
dc.contributor.authorCameron, Peter Jephson
dc.contributor.authorNilson, Tomas
dc.identifier.citationBailey , R A , Cameron , P J & Nilson , T 2018 , ' Sesqui-arrays, a generalisation of triple arrays ' , Australasian Journal of Combinatorics , vol. 71 , no. 3 , pp. 427-451 . < >en
dc.identifier.otherORCID: /0000-0002-8990-2099/work/44629840
dc.identifier.otherORCID: /0000-0003-3130-9505/work/58055606
dc.description.abstractA 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 commonto a row and a column gives a double array. We propose the term sesqui-array for such an array when only the condition on pairs ofcolumns is deleted. Thus all triple arrays are sesqui-arrays.In this paper we give three constructions for sesqui-arrays. The first gives (n+1) x n2 arrays on n(n+1) letters for n>1. (Such an array for n=2 was found by Bagchi.) This construction uses Latin squares.The second uses the 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 x 36 sesqui-array for 42 letters. We also give a construction for K x (K-1)(K-2)/2 sesqui-arrays onK(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 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.
dc.relation.ispartofAustralasian Journal of Combinatoricsen
dc.subjectQA Mathematicsen
dc.titleSesqui-arrays, a generalisation of triple arraysen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.contributor.institutionUniversity of St Andrews. Statisticsen
dc.description.statusPeer revieweden

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