In this talk, I will introduce the concept of POPs and basic concepts regarding avoidance sets, simple permutations and inflation of permutations. I will also demonstrate how to construct explicit bijections from the avoidance sets of certain POPs to a variety of combinatorial objects, and hence address five open questions that were raised in Alice Gao and Sergey Kitaev, On Partially Ordered Patterns of Length 4 and 5 in Permutations, The Electronic Journal Of Combinatorics 26.3 (2019).

Write $\pi = \pi_1 \pi_2\cdots\pi_n$ to denote a permutation $\pi$ on $n$ letters where $\pi(i) = \pi_i$ for $i\in \{1,2,\dots,n\}$. A pattern is a permutation on a set of at least two elements.

A permutation $\pi$ contains a pattern $\sigma:=\sigma_1 \sigma_2 \dots \sigma_\ell$ if there is a subsequence $\pi'=\pi_{i_1}\pi_{i_2}\cdots\pi_{i_\ell}$ (where $1\leq i_1avoids $\sigma$.

We say $\pi$ avoids $\sigma$. For instance, the permutation 4132 contains the pattern 312 (as it has the subsequence 413), while the permutation 1324 avoids the pattern 312. The set of permutations that avoid a pattern or a set of patterns make up an avoidance set.

Enumerating the permutations of given lengths in the avoidance set of a pattern or set of patterns and finding one-to-one correspondences to well-known combinatorial objects is a topic of great interest.

I will explain how the notion of inflation of a permutation was used to successfully attack several such open enumeration questions.

The talk is based on a joint paper with D. Wehlau and I. Zaguia which will appear in Electronic Journal of Combinatorics. A preprint is available on arXiv.

The permutation software PermLab and the Online Encyclopedia of Integer Sequences (OEIS) were very helpful in this work.