Counting Unions of Schreier Sets
Kevin Beanland, Dmitriy Gorovoy, J\c{e}drzej Hodor, Daniil Homza

TL;DR
This paper generalizes the known Fibonacci recurrence for the count of unions of Schreier sets, showing that for any positive integer k, the sequence counting these unions with maximum element n satisfies a linear recurrence.
Contribution
It extends the known Fibonacci recurrence property of Schreier set unions to all positive integers k, establishing a broader linear recurrence relation.
Findings
Sequences $(|(k\mathcal{S})^n|)$ satisfy linear recurrences for all positive k.
Generalization of Fibonacci sequence to broader class of Schreier set unions.
Provides a new understanding of the combinatorial structure of Schreier sets.
Abstract
A subset of positive integers is a Schreier set if it is non-empty and (here is the cardinality of ). For each positive integer , we define as the collection of all the unions of at most Schreier sets. Also, for each positive integer , let be the collection of all sets in with the maximum element equal to . It is well-known that the sequence is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence satisfies a linear recurrence for every positive .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Mathematical Dynamics and Fractals · semigroups and automata theory
