Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.4

On Schreier-Type Sets, Partitions, and Compositions

Kevin Beanland
Department of Mathematics
Washington and Lee University
Lexington, VA 24450

Hùng Việt Chu
Department of Mathematics
Texas A&M University
College Station, TX 77843


For a fixed $\ell\in \mathbb{N}$, a nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geq \ell\vert A\vert-\ell+1$. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in increasing order are non-decreasing. We establish a connection between sparse Schreier-type sets and (restricted) partition numbers. One of our results states that if $\mathcal{G}_{n,\ell}$ consists of partitions of $n$ that contain no parts in $\{2, \ldots, \ell\}$, and we set

$\displaystyle \mathcal{A}_{n,\ell} \ :=\ \{A\subset \{1, \ldots, n\}\,:\, n\in A, A$ is sparse and $\displaystyle \ell$-strong Schreier$\displaystyle \},$    


$\displaystyle \vert\mathcal{A}_{n,\ell}\vert\ =\ \vert\mathcal{G}_{n-1,\ell}\vert,$    for all $\displaystyle n, \ell\in \mathbb{N}.$

The special case $\mathcal{G}_{n-1, 1}$ consists of all partitions of $n-1$. Besides partitions, we also investigate integer compositions.

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(Concerned with sequences A000009 A000041 A000070 A002865 A008483 A008484 A025147 A025148 A025149 A025150 A025151 A027336 A036469 A038348 A185325.)

Received November 6 2023; revised versions received November 8 2023; April 5 2024. Published in Journal of Integer Sequences, April 5 2024.

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