  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.
