论文标题
通过链条和抗小节覆盖物线性扩展的数量的边界
The bounds for the number of linear extensions via chain and antichain coverings
论文作者
论文摘要
令$（\ Mathcal {p}，\ leqslant）$为有限的poset。定义数字$ a_1，a_2，\ ldots $（分别分别$ c_1，c_2，\ ldots $），以便$ a_1+\ ldots+a_k $（分别为$ c_1+\ ldots+c_k $）是$ \ natercal的最大元素$ \ kich $ k的最大元素（然后。推论：如果$ \ MATHCAL {P} $被划分为尺寸$ B_1，B_2，\ LDOTS $的不相交的抗小节，则将$ E（\ Mathcal {p}）\ geqslant \ prod \ prod b_i！$。
Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers $a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$ (respectively, $c_1+\ldots+c_k$) is the maximal number of elements of $\mathcal{P}$ which may be covered by $k$ antichains (respectively, $k$ chains.) Then the number $e(\mathcal{P})$ of linear extensions of poset $\mathcal{P}$ is not less than $\prod a_i!$ and not more than $n!/\prod c_i!$. A corollary: if $\mathcal{P}$ is partitioned onto disjoint antichains of size $b_1,b_2, \ldots$, then $e(\mathcal{P})\geqslant \prod b_i!$.
