Counting Unions of Schreier Sets

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 $F$ is a Schreier set if it is non-empty 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 the maximum element equal to $n$. It is well-known that the sequence $(|(1\mathcal{S})^n|)_{n=1}^\infty$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ satisfies a linear recurrence for every positive $k$.