2022/06/24 by S. V. Avgustinovich, Avgustinovich, Sergey, Sergey Kitaev +3 · 1 citation
Mathematics · #05A05 #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2206.12499
openalex publication_date 2022/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A crucial permutation is a permutation that avoids a given set of prohibitions, but any of its extensions, in an allowable way, results in a prohibition being introduced. In this paper, we introduce five natural types of crucial permutations with respect to monotone patterns, notably quadrocrucial permutations that are linked most closely to Erdős-Szekeres extremal permutations. The way we define right-crucial and bicrucial permutations is consistent with the definition of respective permutations studied in the literature in the contexts of other prohibitions. For each of the five types, we provide its characterization in terms of Young tableaux via the RSK correspondence. Moreover, we use the characterizations to prove that the number of such permutations of length n is growing when n→∞, and to enumerate minimal crucial permutations in all but one case. We also provide other enumerative results.