Some New Congruences for Simultaneously s-Regular and t-Distinct Partition Function

A partition of a positive integer \(n\) is said to be simultaneously \(s\) -regular and \(t\) -distinct partition if none of the parts is divisible by \(s\) and parts appear fewer than \(t\) times. In this paper, we present some new congruences for simultaneously \(s\) -regular and \(t\) -distinct partition function denoted by \(M_{{s,t}}^{d}(n)\) with \((s,t) \in \{ (2,\,\,5),\,\,(3,\,\,4),\,\,(4,\,\,9),\,\,(5\alpha ,\,\,5\beta ),\,\,(7\alpha ,\,\,7\beta ),\,\,(p,\,\,p)\} \) , where \(\alpha \) and \(\beta \) are any positive integers and \(p\) is any prime.