Permutations Avoiding Certain Partially-ordered Patterns

TL;DR

This paper explores the avoidance sets of certain partially ordered patterns in permutations, establishing connections with other combinatorial objects, solving open questions, and developing recursive algorithms for enumeration.

Contribution

It provides new insights into POP avoidance sets, answers open questions, and introduces recursive algorithms and bijections for enumeration and analysis.

Findings

Connected avoidance sets with other combinatorial objects.

Solved five open questions from Gao and Kitaev.

Developed recursive algorithms for enumeration and bijections.

Abstract

A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters are in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev \cite{gao-kitaev-2019}. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive these sets and their corresponding combinatorial objects in parallel, which yielded a natural bijection. We also analysed an avoidance set whose simple permutations are enumerated by the Fibonacci numbers and derived an algorithm to obtain them recursively.