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A. Higgo, A. Kruger, N. Olivier en G. Viljoen (2011-finalejaarstudente, Departement Arbeidsterapie, Universiteit van die Vrystaat) ’n Multiperspektiefbenadering tot die ontwerp van ’n Afrikaanse taakgebaseerde sillabus op universiteitsvlak – Elbie Adendorff: Departement Afrikaans en Nederlands, Universiteit Stellenbosch Gewone lesers aan die Kaap, c. 1680 tot 1850 – Archie L. Dick: Departement van Inligtingkunde, Universiteit van Pretoria Ma’s se persepsies van hoe afknouery gesinne beïnvloed – Estelle Roodt en Abraham P. Greeff: Departement Sielkunde, Universiteit van Stellenbosch Integrasie van artikulatoriese en akoestiese eienskappe van vokale: ’n beskrywingsraamwerk – Daan Wissing: Senior Navorser, Sentrum vir Tekstegnologie, Noordwes-Universiteit Taferele, tableaux vivants, tablo’s en die vroeë Afrikaanse drama (1850–1950) – Marisa Keuris: Departement Afrikaans en Algemene Literatuurwetenskap, Unisa 613 653 686 701 744
LitNet Akademies Jaargang 9 (2), April 2012 Enumerasie van self-ortonogale Latynse vierkante met simmetriese ortogonale maats A.P. Burger, M.P. Kidd en J.H. van Vuuren A.P. Burger, M.P. Kidd en J.H. van Vuuren: Departement Logistiek, Universiteit Stellenbosch Opsomming In hierdie artikel enumereer ons verskeie ekwivalensieklasse van self-ortogonale Latynse vierkante met simmetriese, ortogonale maats (SOLVSOMs), ’n probleem wat nog nie in die literatuur oor kombinatoriese ontwerpe aangespreek is nie. In die besonder bepaal ons die getal (ry, kolom)-paratoopklasse van SOLVSOMs van orde n 10 deur inligting in bestaande, uitputtende databasisse van self-ortogonale Latynse vierkante en simmetriese Latynse vierkante met behulp van ’n boomsoektog met terugkering (Eng: backtracking) te kombineer. Ons bepaal ook die getal verskillende SOLVSOMs, SOLVSOMs in standaardvorm en transponentisomorfismeklasse van SOLVSOMs van ordes n 10 deur gebruikmaking van standaard tegnieke uit abstrakte algebra. In die proses beantwoord ons ’n 34 jaar-oue oop bestaansvraag oor SOLVSOMs van orde 10 deur aan te toon dat geen so ’n ontwerp bestaan nie. Aangesien ’n SOLVSOM van orde n in standaardvorm ekwivalent is aan ’n spelskedule vir ’n gadevermydende gemengde-dubbels rondomtalie-tennistoernooi vir n getroude pare, dui hierdie resultaat daarop dat geen so ’n toernooi vir 10 getroude pare geskeduleer kan word nie. Trefwoorde: Latynse vierkant, self-ortogonale Latynse vierkant, simmetriese Latynse vierkant, SOLVSOM, enumerasie. Abstract Enumeration of self-orthogonal Latin squares with symmetric orthogonal mates A Latin square of order n is an n × n array containing each symbol from a set of n distinct symbols exactly once in every row and every column. We denote the entry in row i and column j of a Latin square L by L(i, j) and take the n symbols from the set Zn = {0,...,n − 1}. The index sets for the rows and columns of a Latin square are also taken as Zn. A Latin square is said to be unipotent if all the entries on its main diagonal are a single symbol from Zn, idempotent if the entries on its main diagonal are all the symbols of Zn in natural order, and reduced if both its first row and first column contain the symbols of Zn in natural order. 1
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