Congruence modulo 4 for Andrews' integer partition with even parts below odd parts

We find and prove a class of congruences modulo 4 for Andrews’ partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{E}\mathcal{O}}(n)$ and further prove that $\overline{\mathcal{E}\mathcal{O}}(n)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.