On avoidance of patterns of the form σ-τ by words over a finite alphabet

Toufik Mansour,Mark Shattuck

doi:10.46298/dmtcs.2140

Abstract

Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for $k$-ary words involving vincular patterns containing a single dash, which explain the majority of the equivalences witnessed for such patterns of length four. When combined with previous results, numerical evidence, and some arguments in specific cases, we obtain the complete Wilf-classification for all vincular patterns of length four containing a single dash. In some cases, our proof shows further that the equivalence holds for multiset permutations since it is seen to respect the number of occurrences of each letter within a word. Some related enumerative results are provided for patterns $&tau;$ of length four, among them generating function formulas for the number of members of [$k$]<sup>$n$</sup> avoiding any $&tau;$ of the form 11$a-b$.

Highlights

The Wilf-classification of patterns is a general question in enumerative combinatorics that has been addressed on several discrete structures mostly in the classical case
Vincular patterns resemble classical patterns, except that some of the letters must be consecutive within an occurrence
Burstein and Mansour [3, 4] considered the Wilf-classification of vincular patterns of length three for k-ary words and found generating function formulas for the number of members of a class in several cases

Summary

Introduction

The Wilf-classification of patterns is a general question in enumerative combinatorics that has been addressed on several discrete structures mostly in the classical case. Burstein and Mansour [3, 4] considered the Wilf-classification of vincular patterns of length three for k-ary words and found generating function formulas for the number of members of a class in several cases. These results have as corollaries the majority of the non-trivial equivalences witnessed for such patterns of length four. We adapt the scanning-elements algorithm described in [7], a technique which has proven successful in enumerating length three pattern avoidance classes for permutations, to the comparable problem involving type (3, 1) vincular patterns and k-ary words

Some General Equivalences

Patterns of length four

Further results

Enumerative results