It is clear that $S(p,q)\subset S(p+q)$. Since a $(p,q)$ shuffle is completely determined by how the $p$ first elements are mapped, the cardinality of $S(p,q)$ is ${p+q\choose p}$. The wedge product of a $p$-form and a $q$-form can be defined as a sum over $(p,q)$ shuffles.