Cycle structure of autotopisms of quasigroups and latin squares

An autotopism of a Latin square is a triple (α, β, γ) of permutations such that the Latin square is mapped to itself by permuting its rows by α, columns by β, and symbols by γ. Let Atp(n) be the set of all autotopisms of Latin squares of order n. Whether a triple (α, β, γ) of permutations belongs to Atp(n) depends only on the cycle structures of α, β, and γ. We establish a number of necessary conditions for (α, β, γ) to be in Atp(n), and use them to determine Atp(n) for n?17. For general n, we determine if (α, α, α)∈Atp(n) (that is, if αis an automorphism of some quasigroup of order n), provided that either αhas at most three cycles other than fixed points or that the non‐fixed points of α are in cycles of the same length. © 2011 Wiley Periodicals, Inc. J Combin Designs     

Further Results on Large Sets of Resolvable Idempotent Latin Squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the off‐diagonal cells can be resolved into disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of RILS(v)s pairwise agreeing on only the main diagonal. In this paper, it is established that there exists an LRILS(v) for any positive integer , except for , and except possibly for .     

Multi-latin squares

A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square.In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable.We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.     

Monogamous latin squares

We show for all n∉{1,2,4} that there exists a latin square of order n that contains two entries γ₁ and γ₂ such that there are some transversals through γ₁ but they all include γ₂ as well. We use this result to show that if n>6 and n is not of the form 2p for a prime p?11 then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of MOLS. Such examples provide pairs of 2-maxMOLS.     

Minimum average congestion of enhanced and augmented hypercubes into complete binary trees

We study the embedding problem of enhanced and augmented hypercubes into complete binary trees. Using tree traversal techniques, we compute the minimum average edge congestion of enhanced and augmented hypercubes into complete binary trees.     

超立方体Q_n的传递剖分

一个图的传递剖分是它的边集的一个划分,且满足图的一个自同构群在其划分后的各个部分组成的集合上作用是传递的.决定了超立方体Q_n的所有G-传递剖分,其中G为Q_n的全自同构群.     

对角方阵的变换群

在不改变对角方阵各行、各列、主对角线、次对角线的元素之集的条件下,其变换群是n次对称群S_n的直积S_n×S_n的子群,因对角拉丁方、对角拉丁方正交侣、幻方、高次幻方、加乘幻方均属此类方阵,本文对构作这类对象及研究它们的计数有重要意义.     

在计算机模拟试验中不同试验设计的比较

计算机模拟试验的设计与建模有许多方法,本文选择了四种不同类型的真模型,用四种试验设计方法,考虑了大小不同的试验方案,并选用了四种建模方法进行比较,给出一些经验性结论,相信这些结论在计算机试验中是有参考价值的。     

The fine structures of three symmetric Latin squares with even orders

Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for any integer n ≥ 5.