Counting crucial permutations with respect to monotone patterns

TL;DR

This paper completes the enumeration of five types of crucial permutations that avoid certain monotone subsequences, advancing understanding of their structure and count.

Contribution

It finalizes the enumeration of all crucial permutations of the five types, building on prior partial results and providing a comprehensive count.

Findings

Complete enumeration of crucial permutations for all lengths and types.

Identification of minimal and next minimal length crucial permutations.

Enhanced understanding of the structure of monotone pattern-avoiding permutations.

Abstract

Recently, Avgustinovich, Kitaev, and Taranenko defined five types of $(k, \ell)-$crucial permutations, which are maximal permutations that do not contain an increasing subsequence of length $k$ or a decreasing subsequence of length $\ell$. Further, Avgustinovich, Kitaev, and Taranenko began the enumeration of the $(k, \ell)-$crucial permutations of the minimal length and the next minimal length and the $(k, 3)-$crucial permutations of all lengths for each of the five types of $(k,\ell)-$crucial permutations. In this paper, we complete the enumeration that Avgustinovich, Kitaev, and Taranenko began.