Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers

A finite nonempty set $F\subset\mathbb{N}$ is Schreier if $\min F\ge |F|$. First, we prove a linear recurrence relation and compute initial counts for Schreier sets consisting of intervals. Two intervals of integers are separated if their union is not an interval. If $\mathcal J_{k,n}$ is the collection of Schreier sets that are the union of exactly $k$ separated intervals, then the sequence $(|\mathcal{J}_{k,n}|)_{n=1}^\infty$ satisfies the characteristic polynomial $p_k(x) = (x-1)^{2k+1}(x+1)^k$. Furthermore, we introduce the new concept of $k$-super-Schreier sets and let $\mathcal{S}_{k,n}$ denote the collection of $k$-super Schreier sets whose maximum is $n$. We show that the sequence $(|\mathcal{S}_{k,n}|)_{n=1}^\infty$ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of $n$.