标号Fano平面的组合复杂性
Combinatorial Intricacies of Labeled Fano Planes
摘要：
给定一个七元集合$X = \{1,2,3,4,5,6,7\}$，有30种方法定义其上的Fano平面。让我们称这样的Fano平面的一条线，也就是说来自$X$的无序三元组为普通的或有缺陷的，根据三元组中两个较小整数之和分别等于或不等于剩余的一个。如果有$s$ { \it defective}线穿过标记Fano平面上的一点，则称该点具有$s$，$0 \leq s \leq 3$阶。在总共35个品系中，9个普通品系属于5种不同的类型，而其余26个缺陷品系产生多达10种不同的类型。它示出，特别是，没有标记的法诺平面可以有零阶的所有点，或功能只是一个点的顺序2。本文还指出了与Steiner三元系中突出构形的联系。
英文摘要：
Given a seven-element set $X = \{1,2,3,4,5,6,7\}$, there are 30 ways to define a Fano plane on it. Let us call a line of such Fano plane, that is to say an unordered triple from $X$, ordinary or defective according as the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of order $s$, $0 \leq s \leq 3$, if there are $s$ {\it defective} lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown, in particular, that no labeled Fano plane can have all points of zeroth order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
