An assortment of problems in permutation patterns: unimodality, equivalence, derangements, and sorting

TL;DR

This paper surveys open problems in permutation patterns, focusing on unimodality, equivalence, derangements, and sorting methods, highlighting key challenges and directions for future research.

Contribution

It compiles and discusses open problems across four major themes in permutation patterns, providing a comprehensive overview for researchers.

Findings

Identifies key open problems in rank-unimodality and Wilf-equivalence.

Highlights the enumeration challenges of derangements in various classes.

Discusses sorting algorithms with restricted operations.

Abstract

We collect open problems in permutation patterns on four themes: rank-unimodality in the permutation pattern poset, Wilf-equivalence and shape-Wilf-equivalence, the enumeration of derangements in permutation classes, and sorting by stacks in series, generalized stacks, and restricted containers (C-machines).