Clusters, generating functions and asymptotics for consecutive patterns in permutations

TL;DR

This paper advances the enumeration and classification of permutations avoiding consecutive patterns using the cluster method, providing new results, generalizations, and confirming several conjectures in the field.

Contribution

It introduces a unified approach to enumerate permutations avoiding patterns, generalizes known results, classifies patterns up to length 6, and proves conjectures about pattern avoidance asymptotics.

Findings

Derived differential equations for pattern occurrence generating functions

Classified all length-6 patterns into equivalence classes

Proved monotone pattern asymptotically easiest to avoid

Abstract

We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function. We also complete the classification of consecutive patterns of length up to 6 into equivalence classes, proving a conjecture of Nakamura. Finally, we show that the monotone pattern asymptotically dominates (in the sense that it is easiest to avoid) all non-overlapping patterns of the same length, thus proving a conjecture of Elizalde and Noy for a positive fraction of all patterns.