,

.

The set of all $(p,q)$ shuffles is denoted by $S(p,q).$

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.