COUNTING UNIONS OF SCHREIER SETS

A subset of positive integers F is a Schreier set if it is nonempty and $|F|\leqslant \min F$ (here $|F|$ 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 maximum element equal to n. It is well known that the sequence $(|(1\mathcal {S})^n|)_{n=1}^\infty $ is the Fibonacci sequence. In particular, the sequence satisfies a linear recurrence. We show that the sequence $(|(k\mathcal {S})^n|)_{n=1}^\infty $ satisfies a linear recurrence for every positive k.