Generating functions and Wilf equivalence for k-embeddings

Noonan and Zeilberger extended the Goulden-Jackson cluster method to find generating functions of word factors, and later, Kitaev, Liese, Remmel, and Sagan found generating functions for word embeddings and proved several results on Wilf equivalence in that setting. We will focus on a new concept called a k-embedding, which encapsulates both word factors and word embeddings, as well as the uninvestigated space between them. We will present some results on generating functions and Wilf equivalence of words allowing k-embeddings, and describe the relationship to previous work in the area of pattern avoidance in words. We will also describe some preliminary results using the cluster method on a particular class of k-embeddings. The work is joint with Brian Miceli and Brian Nakamura.