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

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 $τ$ of length four, among them generating function formulas for the number of members of [$k$]$n$ avoiding any $τ$ of the form 11$a-b$.

Keywords

• pattern avoidance
• vincular patterns
• $k$-ary words
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]