Enumeration schemes for vincular patterns

TL;DR

This paper extends enumeration schemes to vincular patterns, providing algorithms to compute avoidance counts, proving existence under certain conditions, and offering empirical data and conjectures on pattern avoidance classifications.

Contribution

It introduces an algorithm for vincular pattern avoidance enumeration schemes, proves their existence for specific pattern classes, and implements these in Maple with empirical analysis.

Findings

Algorithms for vincular pattern avoidance enumeration schemes

Existence of schemes for certain pattern classes proven

Empirical data and conjectures on Wilf-classification provided

Abstract

We extend the notion of an enumeration scheme developed by Zeilberger and Vatter to the case of vincular patterns (also called "generalized patterns" or "dashed patterns"). In particular we provide an algorithm which takes in as input a set $B$ of vincular patterns and search parameters and returns a recurrence (called a "scheme") to compute the number of permutations of length $n$ avoiding $B$ or confirmation that no such scheme exists within the search parameters. We also prove that if $B$ contains only consecutive patterns and patterns of the form $\sigma_1\sigma_2 ... \sigma_{t-1}-\sigma_t$, then such a scheme must exist and provide the relevant search parameters. The algorithms are implemented in Maple and we provide empirical data on the number of small pattern sets admitting schemes. We make several conjectures on Wilf-classification based on this data. We also outline how to refine schemes to compute the number of $B$-avoiding permutations of length $n$ with $k$ inversions.