论文标题
Steiner系统的组合游戏分布
Combinatorial Game Distributions of Steiner Systems
论文作者
论文摘要
$ \ mathscr {p} $ - 某些组合游戏的位置集具有特殊的组合结构。例如，Conway和Ryba首先调查了Hexad游戏的$ \ Mathscr {p} $ - 是Steiner System $ s（5、6、12）$在洗牌编号中的块集，$ \ MATHCAL {D} _ {d} _ {\ text {sh sh}} $。但是，很少有与Hexad游戏这样的与Steiner系统相关的已知游戏。对于给定的施泰纳系统，我们构建了一个游戏，其$ \ mathscr {p} $ - 位置集是其块集。 通过使用构造游戏，我们获得以下两个结果。首先，我们表征$ \ MATHCAL {D} _ {\ text {sh}} $之间的5040同构$ s（5，6，12）$，带有点集$ \ {0，1，\ ldots，11 \} $。对于每个$ s（5、6、12）$，我们的构造会产生一个游戏，其$ \ mathscr {p} $ - 位置集是其块集。从$ \ Mathcal {d} _ {\ text {sh}} $，我们获得了十六进制游戏，并且该游戏的特征是独特的游戏，在获得的5040游戏中，其位置最少。其次，我们通过使用游戏发行版来表征投影式Steiner三重系统。在这里，Steiner系统的游戏分布$ \ MATHCAL {D} $是从Steiner Systems同构到$ \ Mathcal {d} $获得的游戏中位置数量的频率分布。我们发现，可以将$ s（t，t + 1，v）$的游戏分配分解为对称组件，并且只要且仅当其游戏分配具有独特的对称组件时，steiner三重系统才投射。
The $\mathscr{P}$-position sets of some combinatorial games have special combinatorial structures. For example, the $\mathscr{P}$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, $\mathcal{D}_{\text{sh}}$. There were, however, few known games related to Steiner systems like the hexad game. For a given Steiner system, we construct a game whose $\mathscr{P}$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $\mathcal{D}_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, \ldots, 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $\mathscr{P}$-position set is its block set. From $\mathcal{D}_{\text{sh}}$, we obtain the hexad game, and this game is characterized as a unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system $\mathcal{D}$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $\mathcal{D}$. We find that the game distribution of an $S(t, t + 1, v)$ can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.
