Wilf-equivalence on k-ary words, compositions, and parking functions

In this paper we study pattern-avoidance in the set of words over the alphabet [κ].We say that a word w∈[κ]ⁿcontains a pattern t ∈ [ℓ]^mif ω contains a subsequence order-isomorphic to τ . This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of wordpatterns of length at most six. We also consider analogous problems within the set of integer compositions and the set ofparking functions, which may both be regarded as special types of words, and which contain all permutations. In both theserestricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the fullclassification of these short patterns can be obtained with only a few general bijective arguments, which are applicable topatterns of arbitrary size.