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Sesqui-arrays, a generalisation of triple arrays
University of St Andrews, Scotland.
University of St Andrews, Scotland.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. (Mathematics)
2018 (English)In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 71, no 3, p. 427-451Article in journal (Refereed) Published
Abstract [en]

A 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 rowand a column are (possibly different) non-zero constants. Deleting the condition on the letters common to 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 of columns is deleted. In this paper we give three constructions for sesqui-arrays. Therst gives $(n + 1)\times n^2$ arrays on n(n + 1) letters for $n\geq 2$. (Suchan 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 736 sesqui-array for 42 letters.We also give a construction for K(K-1)(K-2)/2 sesqui-arrays on K(K-1)/2 letters from biplanes. The construction starts with a block of a biplane and produces an array which satises 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.

Place, publisher, year, edition, pages
2018. Vol. 71, no 3, p. 427-451
Keywords [en]
Sesqui array, triple array, biplane
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-30900ISI: 000431776200008Scopus ID: 2-s2.0-85046829751OAI: oai:DiVA.org:miun-30900DiVA, id: diva2:1111699
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Construction methods for triple arraysAvailable from: 2017-06-19 Created: 2017-06-19 Last updated: 2018-07-04Bibliographically approved

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Nilson, Tomas

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