On Five Types of Crucial Permutations with Respect to Monotone Patterns

A crucial permutation is a that avoids given set of prohibitions, but any its extensions, in an allowable way, results prohibition being introduced. 
 In this paper, we introduce five natural types permutations with respect to monotone patterns, notably quadracrucial are linked most closely Erdős-Szekeres extremal permutations. The way define right-crucial and bicrucial consistent the definition respective studied literature contexts other prohibitions. For each types, provide characterization terms Young tableaux via Robinson-Schensted correspondence. Moreover, use characterizations prove number such length $n$ growing when $n\to\infty$, enumerate minimal all one case. We also enumerative results.