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COUNTING UNIONS OF SCHREIER SETS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024
- Published online by Cambridge University Press:
- 27 December 2023, pp. 19-31
- Print publication:
- August 2024
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- Article
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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.
