Fedor Petrov - Steklov Institute of Mathematics at St. Petersburg We discuss several recent inequalities between combinatorial characteristics of posets: hooks and antihooks, chains and antichains, number of linear extensions. Some of them are non-trivial already for Young diagrams. The common tool is majorization theory which I am going to briefly remind. Based on joint works with Igor Pak, Vyacheslav Sokolov and Ivan Bochkov.