Patterns in random permutations avoiding some sets of multiple patterns

TL;DR

This paper studies the distribution of pattern occurrences in random permutations avoiding certain sets of length-3 patterns, revealing asymptotic normality in many cases, contrasting with single-pattern avoidance.

Contribution

It introduces new results on the limiting distribution of pattern counts in permutations avoiding multiple patterns, extending previous single-pattern studies.

Findings

Number of pattern occurrences converges to a limit distribution.

Many cases exhibit asymptotic normality.

Contrasts with known results for single-pattern avoidance.

Abstract

We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.