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Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which countsthe number of partitions of $n$ in which every even part is less than each oddpart. He also defined a partition function $\overline{\mathcal{EO}}(n)$ whichcounts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in whichonly the largest even part appears an odd number of times. Andrews proposed toundertake a more extensive investigation of the properties of$\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families ofcongruences for $\overline{\mathcal{EO}}(n)$. We next study parity propertiesof $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely manyintegers $N$ in every arithmetic progression for which$\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely manyintegers $M$ in every arithmetic progression for which$\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Veryrecently, 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 andHecke eigenforms to prove our results.

On Andrews' integer partitions with even parts below odd parts