Infinite Families of Congruences for Partition Functions ${\bar{\mathfrak{EO}}}$(n) and ${\mathfrak{EO}}_e$(n)

In 2018, Andrews introduced the partition functions ${\mathfrak{EO}}$(n) and ${\bar{\mathfrak{EO}}}$(n). The first of these denotes the number of partitions of n in which every even part is less than each odd part, and the second counts the number of partitions enumerated by the first in which only the largest even part appears an odd number of times. In 2021, Pore and Fathima introduced a new partition function ${\mathfrak{EO}}_e$(n) which counts the number of partitions of n which are enumerated by ${\bar{\mathfrak{EO}}}$(n) together with the partitions enumerated by ${\bar{\mathfrak{EO}}}$(n) where all parts are odd and the number of parts is even. They also proved some particular congruences for ${\bar{\mathfrak{EO}}}$(n) and ${\mathfrak{EO}}_e$(n). In this paper, we establish infinitely many families of congruences modulo 2, 4, 5 and 8 for ${\bar{\mathfrak{EO}}}$(n) and modulo 4 for ${\mathfrak{EO}}_e$(n). For example, if p ≥ 5 is a prime with Legendre symbol $({\frac{-3}{p}})=-1$, then for all integers n ≥ 0 and α ≥ 0, we have ${\bar{\mathfrak{EO}}}(8{\cdot}p^{2{\alpha}+1}(pn+j)+{\frac{19{\cdot}p^{2{\alpha}+2}-1}{3}}){\equiv}0$ (mod 8); 1 ≤ j ≤ (p - 1).