Enumerasie van self-ortogonale Latynse vierkante met simmetriese ortogonale maats
Date
2012-04
Authors
Burger, Alewyn P.
Kidd, Martin P.
Van Vuuren, Jan H.
Journal Title
Journal ISSN
Volume Title
Publisher
LitNet Academic
Abstract
OPSOMMING: 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.
ABSTRACT: 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}g. 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.
ABSTRACT: 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}g. 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.
Description
The original publication is available at http://www.litnet.co.za/
Keywords
Magic squares
Citation
Burger, A.P., Kidd, M.P. & Van Vuuren, J.H. 2012. Enumerasie van self-ortogonale Latynse vierkante met simmetriese ortogonale maats. Litnet Akademies, 9(2):1-24.