Higher Order Fibonacci Sequences from Generalized Schreier sets

Abstract: A Schreier set $S$ is a subset of the natural numbers with $\min S\ge |S|$. It has been known that the sequence $(a_{1,n})$, where $$a_{1,n}\ :=\ |{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S \ge |S|}|,$$ is the Fibonacci sequence. Generalizing this result, we prove that for all $p\in \mathbb{N}$, the sequence $(a_{p,n})$, where $$a_{p, n} \ :=\ |{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S\ge p|S|}|,$$ has a linear recurrence relation of higher order. We investigate further by requiring that ${\rm min}<em>2 S\ge q |S|$, where $\min_2 S$ is the second smallest element of $S$. We prove a linear recurrence relation for the sequence $(a</em>{p, q, n})$, where $$a_{p, q, n} \ :=\ |{S\subseteq \mathbb{N}\,:\,\max S = n, \min S \ge p|S|\mbox{ and } {\rm min}<em>2 S\ge q|S|}|,$$ and discuss a curious relationship between $(a</em>{q, n})$ and $(a_{p, q, n})$.