The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays  

作　　者：Yu-fang ZHANG Jing-yuan CHEN Dian-hua WU Han-tao ZHANG 

机构地区：[1]Department of Mathematics, Guangxi Normal University, Guilin 541004, China [2]School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China [3]Computer Science Department, The University of Iowa, Iowa City, IA 52242, USA

出　　处：《Acta Mathematicae Applicatae Sinica》2018年第4期693-702,共10页应用数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(No.11271089);Guangxi Nature Science Foundation(No.2012GXNSFAA053001);Key Foundation of Guangxi Education Department(No.201202ZD012);Guangxi “Ba Gui” Team for Research and Innovation

摘　　要：Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n（SISORLMA（n） for short） based on N is an n × n array M satisfying the following properties：（1） each row of M is a permutation of N, and at least one column is not a permutation of N;（2） the sums of the n numbers in every row and every column are the same;（3） M is orthogonal to its transpose;（4） the main diagonal and the back diagonal of M are 0, 1, ···, n-1 from left to right. In this paper, it is proved that an SISORLMA（n）exists if and only if n ？ {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ？ {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠2.Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n（SISORLMA（n） for short） based on N is an n × n array M satisfying the following properties：（1） each row of M is a permutation of N, and at least one column is not a permutation of N;（2） the sums of the n numbers in every row and every column are the same;（3） M is orthogonal to its transpose;（4） the main diagonal and the back diagonal of M are 0, 1, ···, n-1 from left to right. In this paper, it is proved that an SISORLMA（n）exists if and only if n ？ {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ？ {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠2.

关 键 词：Diagonally ordered magic square IDEMPOTENT nonelementary rational self-orthogonal row Latinmagic array self-orthogonal Latin squares with holes 

分 类 号：O151.21[理学—数学]