Schreier Multisets and the ss-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 $s\geqslant 2$, as: $F^{(s)}_{2-s} = \cdots = F^{(s)}_0 = 0$, $F^{(s)}_1 = 1$, 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 $n = 1,\ldots, u$. 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.Comment: 11 pages. To appear in Proceedings of the Integers Conference 202

Counting Unions of Schreier Sets

A subset of natural numbers $F$ is called a Schreier set if $|F|\leqslant \min F$ (where $|F|$ is the cardinality of $F$). Let $\mathcal{S}$ denote the family of Schreier sets. Alistair Bird observed that if $\mathcal{S}^n$ denotes all Schreier sets with maximum element $n$, then $(|\mathcal{S}^n|)_{n=1}^\infty$ is the Fibonacci sequence. In this paper, for each $k\in \mathbb{N}$ we consider the family $k\mathcal{S}$, where each set is the union of $k$ many Schreier sets, and prove that each sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ is a linear recurrence sequence and moreover, the recursions themselves can be generated by a simple inductive procedure. Moreover, we develop some more interesting formulas describing the sequence.Comment: comments welcom