Further arithmetic properties of Andrews’ integer partitions with even parts below odd parts

In 2018, Andrews introduced the partition function , which counts the number of partitions of a positive integer n where each even part is less than each odd part. Furthermore, he defined another class of partitions, namely which counts the number of partitions counted by where only the largest even part appears an odd number of times. Thereafter, many mathematicians proved numerous results for . In this paper, we continue the study and prove several congruences modulo 10, 16 and 40 for . We further prove that (10n + 8) and (40n + 38) are almost always divisible by 10 and 40, respectively.