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arXiv · 2101.12061

Permutations Avoiding Certain Partially-ordered Patterns

Kai Ting Keshia Yap·David Wehlau·Imed Zaguia
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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.

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Kai Ting Keshia Yap, David Wehlau, Imed Zaguia. 2021-01-28. Permutations Avoiding Certain Partially-ordered Patterns. https://arxiv.org/abs/2101.12061

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