miun.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf
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
ISI: 000431776200008Scopus ID: 2-s2.0-85046829751OAI: oai:DiVA.org:miun-30900DiVA, id: diva2:1111699
##### Projects
Construction methods for triple arraysAvailable from: 2017-06-19 Created: 2017-06-19 Last updated: 2018-07-04Bibliographically approved

#### Open Access in DiVA

fulltext(250 kB)79 downloads
##### File information
File name FULLTEXT01.pdfFile size 250 kBChecksum SHA-512
1db84f659759b712e160ebbdc4bf5218539693dcbac7d34cd07950825284f171a7c278ad428dd0de2215a45dbfc6172e2a6c24e1124392f2456a16dcb3c0c19d
Type fulltextMimetype application/pdf
Scopus

Nilson, Tomas

#### Search in DiVA

Nilson, Tomas
##### By organisation
Department of Mathematics and Science Education
##### In the same journal
The Australasian Journal of Combinatorics
##### On the subject
Discrete Mathematics

#### Search outside of DiVA

GoogleGoogle Scholar
Total: 79 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available
urn-nbn

#### Altmetric score

urn-nbn
Total: 413 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf