On Andrews' integer partitions with even parts below odd parts

Recently, Andrews defined a partition function 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 (EO) over bar (n) which counts the number of partitions of n enumerated by 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 (EO) over bar (n). In this article, we prove infinite families of congruences for (EO) over bar (n). We next study distribution of (EO) over bar (n). We prove that there are infinitely many integers N in every arithmetic progression for which (EO) over bar (2N) is even; and that there are infinitely many integers M in every arithmetic progression for which (EO) over bar (2M) is odd so long as there is at least one. We further prove that (EO) over bar (n) is even for almost all n. Very recently, Uncu has treated a different subset of the partitions enumerated by (EO) over bar (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. (C) 2020 Elsevier Inc. All rights reserved.