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arXiv 2001.00415math.CO
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Steiner系统的组合对策分布

Combinatorial Game Distributions of Steiner Systems

Yuki Irie

备注：
19 pages, 8 figures
发表信息：
The Electronic Journal of Combinatorics 28(4), (2021), #P4.54
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摘要：

某些组合对策的$\mathscr{P}$ -位置集具有特殊的组合结构。例如，由Conway和Ryba首先研究的十六进制游戏的$\mathscr{P}$ -位置集是Steiner系统$S(5, 6, 12)$在洗牌编号中的块集，$\mathcal{D}_{\text{sh}}$。然而，很少有已知的游戏与斯坦纳系统有关，如hexad游戏。对于给定的Steiner系统，我们构造了一个对策，其$\mathscr{P}$ -位置集就是它的块集.通过构造博弈，我们得到了以下两个结果。首先，我们用点集$\{0, 1, \ldots, 11\}$刻画了5040个同构$S(5, 6, 12)$中的$\mathcal{D}_{\text{sh}}$。对于每个$S(5, 6, 12)$，我们的构造产生一个博弈，其$\mathscr{P}$ -位置集是它的块集。从$\mathcal{D}_{\text{sh}}$，我们得到了六角博弈，并且这个博弈被刻画为在所得到的5040个博弈中位置数最少的唯一博弈。其次，我们利用博弈分布刻画了投射Steiner三元系。这里，Steiner系统$\mathcal{D}$的博弈分布是从同构于$\mathcal{D}$的Steiner系统获得的博弈中的位置数的频率分布。我们发现$S(t, t + 1, v)$的博弈分布可以分解成对称分量，并且Steiner三元系是投射的当且仅当它的博弈分布有唯一的对称分量.

英文摘要：

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.

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。

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