On five types of crucial permutations with respect to monotone patterns

TL;DR

This paper introduces five types of crucial permutations related to monotone patterns, characterizes them via Young tableaux, and studies their enumeration and growth properties.

Contribution

It defines and characterizes five natural types of crucial permutations with respect to monotone patterns, linking them to Young tableaux and enumerating their counts.

Findings

Number of such permutations grows with length n

Explicit enumeration of minimal crucial permutations in most cases

Characterizations via RSK correspondence

Abstract

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 Erd\H{o}s-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 $n\to\infty$, and to enumerate minimal crucial permutations in all but one case. We also provide other enumerative results.