链与反链的真归并计数
Counting Proper Mergings of Chains and Antichains
摘要：
两个不相交的拟序集$P$和$Q$的适当合并是$P$和$Q$的并集上的拟序，使得对$P$和$Q$的限制再次产生原始拟序，并且使得$P$和$Q$中没有元素被标识。在这篇文章中，我们考虑的情况下，其中$P$和$Q$是链，其中$P$和$Q$是反链，其中$P$是反链和$Q$是链。我们给出的公式，确定在所有三种情况下的适当合并的数量，并引入两个新的双射从适当合并的两个链的平面分区和适当合并的反链和链的单调染色的完全二部有向图。此外，我们还利用这些双射分别计算了两条链之间以及链与布尔格之间的伽罗瓦连接数。
英文摘要：
A proper merging of two disjoint quasi-ordered sets $P$ and $Q$ is a quasi-order on the union of $P$ and $Q$ such that the restriction to $P$ and $Q$ yields the original quasi-order again and such that no elements of $P$ and $Q$ are identified. In this article, we consider the cases where $P$ and $Q$ are chains, where $P$ and $Q$ are antichains, and where $P$ is an antichain and $Q$ is a chain. We give formulas that determine the number of proper mergings in all three cases, and introduce two new bijections from proper mergings of two chains to plane partitions and from proper mergings of an antichain and a chain to monotone colorings of complete bipartite digraphs. Additionally, we use these bijections to count the Galois connections between two chains, and between a chain and a Boolean lattice respectively.
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。
