Emerging consecutive pattern avoidance
Nathanaël Hassler, Sergey Kirgizov
TL;DR
This work addresses the asymptotic popularity of length-3 consecutive patterns in eighteen permutation-avoidance classes identified by Kitaev and Mansour. It primarily uses a mix of analytic generating-function techniques and bijective mappings (notably to involutions via the Foata transform) to derive exact asymptotic frequencies for many classes, while exploiting symmetry under $R$, $C$, and $R\circ C$. For simple classes, the limiting pattern frequencies follow directly from structural restrictions; for the nontrivial classes, the authors obtain precise values such as $pop_{11}(231)=\tfrac{1}{2}$ and $pop_{11}(213)=pop_{11}(312)=\tfrac{1}{4}$ in Av$_n$(123,132,321), and $pop_{17}(321)=0$, $pop_{17}(231)=\tfrac{1}{2}$, $pop_{17}(213)=pop_{17}(312)=\tfrac{1}{4}$ in Av$_n$(123,132). The paper also presents a conjecture about the invariance of asymptotic frequencies when adding a pattern to an avoidance class and outlines open questions for several unresolved classes. Overall, the results reveal that certain consecutive patterns can vanish asymptotically in these restricted permutation classes, and the methods blend generating-function analysis with combinatorial bijections to obtain sharp limits.
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.