Wilf Equivalence in Interval Embeddings

Consider the alphabet A and define A∗ as the set of words over A. Define a vector of sequences of subsets of N as ~u = (u1, u2, . . . , uk). Consider a word w ∈ A∗. Define their to be an embedding of ~u in w, ~u ≤ w if there is some i such that, wi ∈ uj , wi+1 ∈ uj+1, . . . wi+k−1 ∈ uj+k−1. Define a word that avoids the vector ~u as a word where there is no such i, such that wi ∈ uj , wi+1 ∈ uj+1, . . . wi+k−1 ∈ uj+k−1. We define the weight of a function as wt(w) = t|w|x ∑ (w). We define the generating function for a certain pattern ~u as F (u;x, t) = ∑ u≤w wt(w). We consider two patterns ~u and ~v to be Wilf Equivalent if F (~u;x, t) = F (~v;x, t). We then prove some properties for Wilf Equivalence of patterns. We use these properties to then try to describe classes of Wilf Equivalent objects.