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: $F_{2-s}^{(s)}=\cdots =F_0^{(s)}=0$, $F_1^{(s)}=1$, and $F^{(s)}_{n} = F^{(s)}_{n-1} + \cdots + F^{(s)}_{n-s}, \mbox{ for } n\geqslant 2$. 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 $n=1,\dots ,u$. 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.

11 pages. To appear in Proceedings of the Integers Conference 2023