Counting Unions of Schreier Sets

A subset of positive integers $F$ is a Schreier set if it is non-empty and $\mathrm{\mid }F\mathrm{\mid }⩽\min F$ (here $\mathrm{\mid }F\mathrm{\mid }$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most $k$ Schreier sets. Also, for each positive integer $n$, let $(k\mathcal{S}{)}^n$ be the collection of all sets in $k\mathcal{S}$ with the maximum element equal to $n$. It is well-known that the sequence $(\mathrm{\mid }(1\mathcal{S}{)}^n\mathrm{\mid }{)}_{n=1}^{\mathrm{\infty }}$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(\mathrm{\mid }(k\mathcal{S}{)}^n\mathrm{\mid }{)}_{n=1}^{\mathrm{\infty }}$ satisfies a linear recurrence for every positive $k$.