论文标题
在代表子系统分区的拉丁广场的国际象棋板上分布。
Distribution of rooks on a chess-board representing a Latin square partitioned by a subsystem
论文作者
论文摘要
A $ D $维订单$ n $的拉丁广场的概括可以被视为尺寸$ n \ times n \ times n \ times \ ldots \ times \ times n $（$ d $ times）的棋盘，其中包含$ n^{d-1} $ n^{d-1} $ nontabcking roks。每个单元格由$ d $ -tuple $（e_1，e_2，\ ldots，e_d）$识别，其中$ e_i \ in \ {1,2，\ ldots，n \} $。对于$ d = 3 $，我们证明了这样的国际象棋板代表一个主要类。 一个子系统$ t $由$ <e_1，e_2，\ ldots，e_d> $超过$ \ {1,2，\ ldots，n \} $的子系统是真实的。 $ t $的密度是包含的Rooks与$ t $中的单元数的比例。两个子系统之间的距离是细胞对之间的最小锤距。用$ <e_1，e_2，\ ldots，e_d> $替换$ k $ sets，用$ k $在$ t $和$ u $之间获得子系统$ u $。所有这些子系统，包括$ t $，构成了国际象棋棋盘的分区。我们证明，在这样的分区中，可以从$ t $中的新闻数量和$ t $和$ u $的单元格数量以及$（1）^k $的值确定$ u $的Rooks数量和$ U $的密度。我们检查了$ 2 $ - 和$ 3 $维的案例中的子系统夫妇$（t，u）$，其中$ u $是与实际$ t $的最远的独特子系统。在即时证明了二项式系数的新身份。
A $d$-dimensional generalization of a Latin square of order $n$ can be considered as a chess-board of size $n\times n\times \ldots\times n$ ($d$ times), containing $n^d$ cells with $n^{d-1}$ non-attacking rooks. Each cell is identified by a $d$-tuple $(e_1,e_2,\ldots ,e_d)$ where $e_i \in \{1,2,\ldots ,n\}$. For $d = 3$ we prove that such a chess-board represents precisely one main class. A subsystem $T$ induced by a family of sets $<E_1,E_2,\ldots ,E_d>$ over $\{1,2,\ldots ,n\}$ is real if $E_i \subset \{1,2,\ldots ,n\}$ for each $i \in \{1,2,\ldots ,d\}$. The density of $T$ is the ratio of contained rooks to the number of cells in $T$. The distance between two subsystems is the minimum Hamming distance between cell pairs. Replacing $k$ sets of $<E_1,E_2,\ldots ,E_d>$ by their complements, a subsystem $U$ is obtained with distance $k$ between $T$ and $U$. All these subsystems, including $T$, form a partition of the chess-board. We prove that in such a partition, the number of rooks in a $U$ and the density of $U$ can be determined from the number of rooks in $T$ and the number of cells in $T$ and $U$ and the value of $(-1)^k$. We examine the subsystem couple $(T,U)$ in the $2$- and $3$-dimensional cases, where $U$ is the most distant unique subsystem from a real $T$. On the fly, a new identity of binomial coefficients is proved.
