Congruences for Simultaneously s-Regular, t-Regular and s-Distinct Partition Function

Abstract

Keith (Integers 23, A9, 2023) introduced the simultaneously s-regular, t-regular and s-distinct partition function which counts the total number of partitions of a positive integer n such that none of the parts are divisible by s and t and each part appears fewer than s times. The simultaneously s-regular, t-regular and s-distinct partition function is denoted by $B_{s,t}^D(n)$, where $1<s<t$ are integers. In this paper, we prove some infinite families of congruences for the partition function $B_{s,t}^D(n)$ for $(s,t)=(3,4)$, (4, 9), (5x, 5y) and (7x, 7y), where x and y are two positive integers. We also offer congruences of $B_{s,t}^D(n)$ for prime values of s.