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.