Schreier Multisets and the S-Step Fibonacci Sequences

Abstract

Inspired by the surprising relationship (due to A. Bird) between Schreier sets and the Fibonacci sequence, we introduce Schreier multisets and connect these multisets with the s-step Fibonacci sequences, defined, for each s ≥ 2, as: F(s) 2−s=· · ·= F(s)0 = 0, F(s)1 = 1, and Fn(s) = F(s)n−1+· · ·+Fn−s,(s) for n ≥ 2. Next, we use Schreier-type conditions on multisets to retrieve a family of sequences which satisfy a recurrence of the form a(n) = a(n − 1) + a(n − u), with a(n) = 1 for n = 1, …, u. Finally, we study nonlinear Schreier conditions and show that these conditions are related to integer decompositions, each part of which is greater than the number of parts raised to some power. © 2024, Colgate University. All rights reserved.