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We find and prove a class of congruences modulo 4 for Andrews' partition withcertain ternary quadratic form. We also discuss distribution of$\overline{\mathcal{EO}}(n)$ and further prove that$\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study wasinspired by similar congruences modulo 4 in the work by the second author andGarvan.

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