Pattern avoidance and the fundamental bijection

2024/07/08 by Archer, Kassie, Laudone, Robert P. · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.06338

Abstract

The fundamental bijection is a bijection θ:S_n\toS_n in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations π∈ S_n that avoids a pattern σ∈ S₃, whose image θ(π) also avoids σ. We additionally consider what happens under repeated iterations of θ; in particular, we enumerate permutations π∈ S_n that have the property that π and its first k iterations under θ all avoid a pattern σ. Finally, we consider permutations with the property that π=θ²(π) that avoid a given pattern σ, and end the paper with some directions for future study.

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• Characterizing avoidance in cycles via vincular patterns

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