On Schreier-Type Sets, Partitions, and Compositions

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 $\displaystyle \},$

then

$\displaystyle \vert\mathcal{A}_{n,\ell}\vert\ =\ \vert\mathcal{G}_{n-1,\ell}\vert,$ 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.

On Schreier-Type Sets, Partitions, and Compositions

Kevin Beanland Department of Mathematics Washington and Lee University Lexington, VA 24450 USA Hùng Việt Chu Department of Mathematics Texas A&M University College Station, TX 77843 USA

Kevin Beanland
Department of Mathematics
Washington and Lee University
Lexington, VA 24450
USA
Hùng Việt Chu
Department of Mathematics
Texas A&M University
College Station, TX 77843
USA