Schreier Multisets and the $s$-step Fibonacci Sequences

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 $F_n^{(s)}=F_{n-1}^{(s)}+\cdots +F_{n-s}^{(s)},forn⩾2$. 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.