论文标题
交替的符号超质子分解拉丁正方形
Alternating Sign Hypermatrix Decompositions of Latin-like Squares
论文作者
论文摘要
对于任何$ n \ times n $ latin square $ l $，我们可以将相互正交置换矩阵的唯一顺序关联，$ p = p_1，p_1，p_2，...，p_n $，这样$ l = l = l = l（p）= \ sum kp_k $。 Brualdi和Dahl（2018）描述了拉丁广场的概括，称为交替标志Hypermatrix Like Square（Ashl），用交替的标志Hypermatrix（ASHM）代替$ p $。 ASHM是$ n \ times n \ times n $（0,1，-1）-Hypermatrix，其中每行，列和垂直线在标志中的非零元素在符号中替代，以$ 1 $开始和结尾。由于$ n $相互正交的置换矩阵的每个顺序构成了独特的$ n \ times n \ times n $ ashm的平面，因此这种拉丁平方的概括非常自然地遵循，而Ashm $ a $具有相应的Ashl $ l = l = l = l = l = l = \ sum ka_k $，$ a_k $，$ a_k $ a $ a_k $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $本文解决了Brualdi和Dahl的文章中提出的一些空旷的问题，首先是表征与相同的Ashl对相对应相互关系的Ashm，并在$ n $上提供了一个紧密的下下限，而两个$ n \ times n \ times n \ times n $ ashms可以对应于同一ashl，其次可以通过与nite n osher n osh n osh n osh n osh n osh n osh n osh n osh n osh n oshl一起探索。给出了$ n \ times n $ ashl的一般结构，同一条目发生了$ \ lfloor \ frac {n^2 + 4n -19} {2} {2} \ rfloor $ times，从而在以前的最佳结构上有了很大的改进，从而实现了同一条目，该条目实现了$ 2n $ times的同一条目。
To any $n \times n$ Latin square $L$, we may associate a unique sequence of mutually orthogonal permutation matrices $P = P_1, P_2, ..., P_n$ such that $L = L(P) = \sum kP_k$. Brualdi and Dahl (2018) described a generalisation of a Latin square, called an alternating sign hypermatrix Latin-like square (ASHL), by replacing $P$ with an alternating sign hypermatrix (ASHM). An ASHM is an $n \times n \times n$ (0,1,-1)-hypermatrix in which the non-zero elements in each row, column, and vertical line alternate in sign, beginning and ending with $1$. Since every sequence of $n$ mutually orthogonal permutation matrices forms the planes of a unique $n \times n \times n$ ASHM, this generalisation of Latin squares follows very naturally, with an ASHM $A$ having corresponding ASHL $L = L(A) =\sum kA_k$, where $A_k$ is the $k^{\text{th}}$ plane of $A$. This paper addresses some open problems posed in Brualdi and Dahl's article, firstly by characterising how pairs of ASHMs with the same corresponding ASHL relate to one another and providing a tight lower bound on $n$ for which two $n \times n \times n$ ASHMs can correspond to the same ASHL, and secondly by exploring the maximum number of times a particular integer may occur as an entry of an $n \times n$ ASHL. A general construction is given for an $n \times n$ ASHL with the same entry occurring $\lfloor\frac{n^2 + 4n -19}{2}\rfloor$ times, improving considerably on the previous best construction, which achieved the same entry occuring $2n$ times.
