Consecutive and quasi-consecutive patterns: $\mathrm{des}$-Wilf classifications and generating functions

TL;DR

This paper classifies quasi-consecutive permutation patterns of length up to 4 based on descent distribution, establishing equivalences via bijections and deriving explicit generating functions for specific classes.

Contribution

It provides a complete descent-Wilf classification for small quasi-consecutive patterns and corrects previous incomplete bijective proofs with new bijections and generating functions.

Findings

Complete classification for patterns of length up to 4

Construction of descent-preserving bijections

Explicit generating functions for certain classes

Abstract

Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as established in this paper, we obtain a complete $\des$-Wilf classification for quasi-consecutive patterns of length up to 4. For equivalence classes containing more than one pattern, we construct various descent-preserving bijections to establish the equivalences, which lead to the provision of proper versions of two incomplete bijective arguments previously published in the literature. Additionally, for two singleton classes, we derive explicit bivariate generating functions using the generalized run theorem.