Avoidance of Partially Ordered Generalized Patterns of the form $k$-$\sigma$-$k$

TL;DR

This paper extends the understanding of pattern avoidance in permutations by relating exponential generating functions for avoiding complex generalized patterns, and provides bijections and classifications for specific pattern avoiders.

Contribution

It generalizes previous results to multiple patterns and partially ordered patterns, offering new formulas, bijections, and classifications in permutation pattern avoidance.

Findings

Derived exponential generating functions for avoiding multiple generalized patterns.

Constructed a bijection between bicolored set partitions and pattern-avoiding permutations.

Provided a complete classification of avoidance sets for single partially ordered patterns.

Abstract

Sergey Kitaev has shown that the exponential generating function for permutations avoiding the generalized pattern $\sigma$-$k$, where $\sigma$ is a pattern without dashes and $k$ is one greater than the biggest element in $\sigma$, is determined by the exponential generating function for permutations avoiding $\sigma$. We show that this also holds for permutations avoiding all the generalized patterns $\sigma_1$-$k_1$, $...$, $\sigma_n$-$k_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the biggest element in $\sigma_i$. Similarly the exponential generating function for permutations avoiding the partially ordered generalized patterns $k_1$-$\sigma_1$-$k_1$, $...$, $k_n$-$\sigma_n$-$k_n$ can be determined from the exponential generating function for permutations avoiding the generalized patterns $\sigma_1$, $...$, $\sigma_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the largest element in $\sigma_i$. Using this we construct a bijection between bicolored set partitions and permutations avoiding the partially ordered generalized pattern 3-12-3 (that is, permutations avoiding both the patterns 3-12-4 and 4-12-3). By using this method twice, we find a closed formula for the exponential generating function for permutations avoiding the partially ordered generalized pattern 3-121-3. Finally, we give a complete classification of when single partially ordered generalized patterns have the same set of avoiders.