Generating functions and Wilf equivalence for generalized interval embeddings

In 1999 in [J. Difference Equ. Appl. 5, 355–377], Noonan and Zeilberger extended the Goulden-Jackson Cluster Method to find generating functions of word factors. Then in 2009 in [Electron. J. Combin. 16(2), RZZ], Kitaev, Liese, Remmel and Sagan found generating functions for word embeddings and proved several results on Wilf-equivalence in that setting. In this article, the authors focus on generalized interval embeddings, which encapsulate both factors and embeddings, as well as the “space between” these two ideas. The authors present some results in the most general case of interval embeddings. Two special cases of interval embeddings are also discussed, as well as their relationship to results in previous works in the area of pattern avoidance in words.