Recently, Andrews introduced partition functionsEO(n) and EO¯(n) where the function EO(n) denotes the number of partitions of n in which every even part is less than each odd part and the function EO¯(n) denotes the number of partitions enumerated by EO(n) in which only the largest even part appears an odd number of times. In this paper we obtain some congruences modulo 2, 4, 10 and 20 for the partition function EO¯(n). We give a simple proof of the first Ramanujan-type congruences EO¯10n+8≡0 mod5 given by Andrews.

Keywords: partitions, partitions with even parts below odd parts, congruences.