Characterizing avoidance in cycles via vincular patterns

2025/05/08 by Robert P. Laudone, Laudone, Robert P.
Biochemistry, Genetics and Molecular Biology · Mathematics · #05A05 #05A15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Genome Rearrangement Algorithms #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2505.05651

openalex publication_date 2025/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that cyclic permutations avoiding 321 are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of 321 avoidance in terms of equality of the upper bound of the Daiconis-Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of 321 avoiding cycles.

Citations

• Pattern avoidance and the fundamental bijection
• Pattern-functions, statistics, and shallow permutations
• Cyclic permutations realized by signed shifts
• Simple permutations and pattern restricted permutations
• Permutations with restricted patterns and Dyck paths

Related

• Cyclic Pattern Containment and Avoidance
• Binary operations on pattern-avoiding cycles
• Pattern avoidance and the fundamental bijection
• Pattern-restricted cyclic permutations with a pattern-restricted cycle form
• Pattern-avoiding shallow permutations