A Complete Enumeration of Ballot Permutations Avoiding Sets of Small Patterns

TL;DR

This paper fully enumerates ballot permutations avoiding sets of small patterns, establishes recurrence relations, and finds bijections with Dyck path factors, advancing understanding of pattern avoidance in ballot permutations.

Contribution

It provides a complete enumeration and classification of ballot permutations avoiding multiple length-3 patterns, including new bijections and recurrence formulas.

Findings

Enumeration of ballot permutations avoiding two length-3 patterns

Recurrence relations and formulas for avoidance classes

Bijection between ballot permutations avoiding 132 and 312 with Dyck path factors

Abstract

Permutations whose prefixes contain at least as many ascents as descents are called ballot permutations. Lin, Wang, and Zhao have previously enumerated ballot permutations avoiding small patterns and have proposed the problem of enumerating ballot permutations avoiding a pair of permutations of length $3$. We completely enumerate ballot permutations avoiding two patterns of length $3$ and we relate these avoidance classes with their respective recurrence relations and formulas, which leads to an interesting bijection between ballot permutations avoiding $132$ and $312$ with left factors of Dyck paths. In addition, we also conclude the Wilf-classification of ballot permutations avoiding sets of two patterns of length $3$, and we then extend our results to completely enumerate ballot permutations avoiding three patterns of length $3$.