Schreier-Type Sets and Linear Recurrences: Connections and Developments

We demonstrate several common techniques for proving linear recurrences from counting Schreier-type sets. These techniques include formula-based arguments, bijective proofs, mathematical induction, the inclusion-exclusion principle, and the characteristic polynomial method. As new contributions, we examine symmetric maximal Schreier sets, Schreier sets that contain a prescribed integer, and Schreier sets that avoid integers belonging to a fixed arithmetic progression. Along the way, we employ useful techniques for identifying meaningful patterns in data and establishing technical identities. The results presented here, together with the diverse proof techniques employed, are expected to serve as a valuable resource for undergraduate researchers interested in this area.