On Schreier-type Sets, Partitions, and Compositions

A nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geqslant \ell|A|-\ell+1$. 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. This note establishes 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 \begin{equation*} \mathcal{A}_{n,\ell} \ :=\ \{A\subset \{1, \ldots, n\}\,:\, n\in A, A\mbox{ is sparse and }\ell\mbox{-strong Schreier}\}, \end{equation*} then $$|\mathcal{A}_{n,\ell}|\ =\ |\mathcal{G}_{n-1,\ell}|, \quad n, \ell\in \mathbb{N}.$$ The special case $\mathcal{G}_{n-1, 1}$ consists of all partitions of $n-1$. Besides partitions, integer compositions are also investigated.