用正交拉丁方构造两次幻方

起源于我国的幻方,自从费尔马提出幻立方的概念后,研究者多向高维方面发展。作者在[6]—[12]中曾探讨过幻方和幻立方的平方和相等性问题。本文提出了一个新的概念:两次幻方,给出了构成两次幻方的充分条件,并提供了一个构造2~m阶和(2m+1)~2阶两次幻方的方法。     

4n阶标准二次幻方的构作

给出标准二次幻方及等重集的概念.利用2n阶正交截态拉丁方,Z4n={0,1,…,4n-1}的对称2次等幂和等分（划分）以及方阵的简单变换构作了4n（n≥2,n≠3）阶标准二次幻方.由于n=3时,存在12阶标准二次幻方,而n=1时,不存在4阶标准二次幻方,故4n阶标准二次幻方的存在性已经完全解决.     

强对称的自正交对角拉丁方

1 引言 一个ｎ阶拉丁方是含ｎ个相异元素的集合Ｎ上的一个ｎ阶方阵，其每一行和每一列都是Ｎ的一个置换．ｎ阶拉丁方的一条截态是位于不同行不同列的ｎ个位置使得其中的ｎ个元素两两相异．ｎ阶对角拉丁方是一个ｎ阶拉丁方，其主对角线（位置（）与反对角线（位置（）均为截态． 两个ｎ阶拉丁方Ａ和Ｂ称为正交的（简记作Ａ上Ｂ），如果把它们迭合在一起时，拉丁方Ａ的每一个记号与拉丁方Ｂ的每一个记号相遇一次且仅相遇一次．如果一个ｎ阶拉丁方Ｌ和它自己的转置正交，则称Ｌ为一个自正交的拉丁方，简记为ＳＯＬＳ（ｎ）． ｎ阶自正交对角拉…     

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.     

用正交对角拉丁方及幻矩与和阵构作mn阶标准二次幻方

当m和n为同奇或同偶的正整数且m,n≠1,2,3,6时,用m和n阶正交对角拉丁方及{0,1,…,mn-1）上的m×n幻矩与和阵,构作了mn阶标准二次幻方．     

一种4N阶幻方构造方法的论证

一种 4 N阶幻方构造方法被发现 .本文阐明了 4 N阶幻方构造方法 ,介绍了 12阶幻方构造过程 .     

正交拉丁方组和泛对角线幻方的一种构造法

利用线性取余变换构造素数阶完备正交拉丁方组,给出泛对角线幻方的一种构造法.     

最佳拉丁方与高级原幻方

本文证明了 (n ,2 ) =(n ,3) =1时 ,有n阶的正交的最佳拉丁方。若n =4k ,或n是个不为 3的奇数 ,则有n阶的正交的高级原幻方     

用线性取余变换造正交拉丁方和幻方

本文利用线性取余变换造正交拉丁方、幻方和泛对角线幻方。文［１］造奇数阶正交拉丁方的方法，文［２］的方法都本文方法的特例。     

三次幻方的构作

给出2k维m阶t次幻方及m模方阵,m模列满秩矩阵,模线,m经典模线集和t次m模基因阵的概念,并用矩阵法和组合法初步研究了t次幻方特别是三次幻方的构作.证明:(i)若存在2k阶t次m模基因阵,则存在2k维m阶t次幻方;(ii)若N=P1α1P2α2…PSαS为N的标准分解式,iα≥3,Piiα≥16(1≤i≤S),则存在二维N阶三次幻方;(iii)若存在二维偶m阶2t+1次幻方和二维n阶2t次幻方,则存在二维mn阶2t+1次幻方;(iv)若存在二维m阶和n阶t次幻方,则存在二维mn阶t次幻方;(v)当t≥3时,不存在二维单偶数阶t次幻方.     

Orthogonal factor‐pair Latin squares of prime‐power order

We introduce a linear method for constructing factor‐pair Latin squares of prime‐power order and we identify criteria for determining whether two factor‐pair Latin squares constructed using this linear method are orthogonal. Then we show that families of pairwise mutually orthogonal diagonal factor‐pair Latin squares exist in all prime‐power orders.     

Existence of (3, 1, 2)‐conjugate orthogonal diagonal latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3,1,2)‐conjugate, and the latter is also a diagonal Latin square, as a (3,1, 2)‐conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers υ except 2, 3, 6, and possibly 10. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 297–308, 2001     

Decomposition of Bicolored Square Arrays into Bichromatic Diagonals

Let M be an array (), where each of its cells is colored in one of two colors. We give a necessary and sufficient condition for the existence of a partition of M into n diagonals, each containing at least one cell of each color. As a consequence, it follows that if each color appears in at least cells, then such a partition exists. The proof uses results on completion of partial Latin squares.     

When is a partial Latin square uniquely completable, but not its completable product?

Let P and Q be uniquely completable partial Latin squares. It is an open problem to determine necessary and sufficient conditions so that the completable product P⊗Q is also uniquely completable. So far, only a few specific examples of P have been given such that the completable product of P with itself (P⊗P) does not have a unique completion. In this paper, we find a whole class of such partial Latin squares.     

D[X2m]的下界估计与X2m矩阵的快速构造

本文利用有向图理论研究了Xn矩阵的特性，给出了X2m矩阵类的快速构造方法，证明了拉丁方矩阵D[X2m]数目的下界估计是：2m(2m)! ∑^mi=2[(2m)!]^2/П^ij=1Kj!П^rj=1bj!。     

A short disproof of Euler’s conjecture based on quasi-difference matrices and difference matrices

In this note, two classes of quasi-difference matrices, $(2n+2,4;1,1;n)$-QDM and $(4n+1,4;1,1;2n?1)$-QDM, are constructed. Combining the known results of quasi-difference matrices and difference matrices, a new short disproof of Euler’s conjecture on mutually orthogonal Latin squares is given.     

A graph-theoretic proof of the non-existence of self-orthogonal Latin squares of order 6

The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given, but these proofs generally require extensive prior subject knowledge in order to follow them, or are ‘blind’ proofs in the sense that most of the work is done by computer or by exhaustive enumeration. In this paper we present a graph-theoretic proof of a somewhat weaker result, namely the non-existence of self-orthogonal Latin squares of order six, by introducing the concept of a self-orthogonal Latin square graph. The advantage of this proof is that it is easily verifiable and accessible to discrete mathematicians not intimately familiar with the theory of combinatorial designs. The proof also does not require any significant prior knowledge of graph theory.     

完美置换和空间均衡拉丁方北大核心CSCD

本文首次提出完美置换的概念并研究它的代数性质和构造方法,解决了2n+1为素数时n阶完美置换的存在性.我们还利用完美置换给出了循环空间均衡拉丁方和对称空间均衡拉丁方的构造方法,它们在试验设计中有广泛的应用。