Permutations avoiding certain partially-ordered patterns (Q2048546)

Summary: A permutation \(\pi\) contains a pattern \(\sigma\) if and only if there is a subsequence in \(\pi\) with its letters 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 \textit{A. L. L. Gao} and \textit{S. Kitaev} in [Electron. J. Comb. 26, No. 3, Research Paper P3.26, 31 p. (2019; Zbl 1418.05006)]. 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 natural bijections. We also analysed an avoidance set whose simple permutations are enumerated by the Fibonacci numbers and derived an algorithm to obtain them recursively.