We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the inequalities, and the computational complexity aspects of the results.

翻译：我们全面综述了有限偏序集线性扩展数量的不等式。我们回顾了大量实例，讨论了未决问题，并介绍了该主题的最新研究成果。我们重点强调不等式的界限、等号成立条件以及结果的计算复杂性方面。