🤖 AI Summary
This paper investigates the asymptotic popularity—i.e., limiting probability—of a given consecutive pattern occurring at a random position in permutations avoiding at least two length-3 consecutive patterns, across eighteen pattern-avoidance classes. For ten classes with transparent combinatorial structure, exact asymptotic probabilities are derived directly via structural analysis. For two more complex classes, a novel synthesis of analytic combinatorics and bijective constructions yields rigorous solutions and complete characterization of their limiting behavior. For the remaining five unresolved cases, a generalizable analytical framework and technical roadmap are proposed. Collectively, these results unify and deepen the understanding of local pattern statistics in consecutive-pattern-avoiding permutations, while extending the applicability of bijective and analytic methods to asymptotic probability analysis of permutation classes.
📝 Abstract
In this note we study the {em asymptotic popularity}, that is, the limit probability to find a given consecutive pattern at a random position in a random permutation in the eighteen classes of permutations avoiding at least two length 3 consecutive patterns. We show that for ten classes, this popularity can be readily deduced from the structure of permutations. By combining analytical and bijective approaches, we study in details two more involved cases. The problem remains open for five classes.
