  Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts
the number of partitions of $n$ in which every even part is less than each odd
part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which
counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which
only the largest even part appears an odd number of times. Andrews proposed to
undertake a more extensive investigation of the properties of
$\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of
congruences for $\overline{\mathcal{EO}}(n)$. We next study parity properties
of $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely many
integers $N$ in every arithmetic progression for which
$\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely many
integers $M$ in every arithmetic progression for which
$\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Very
recently, Uncu has treated a different subset of the partitions enumerated by
$\mathcal{EO}(n)$. We prove that Uncu's partition function is divisible by
$2^k$ for almost all $k$. We use arithmetic properties of modular forms and
Hecke eigenforms to prove our results.
