$(p,q)$ shuffle

Definition.

Let $p$ and $q$ be positive natural numbers. Further,let $S\mathit{}\mathrm{(}k\mathrm{)}$ be the set of permutations of the numbers $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}k\mathrm{\}}$.A permutation $\tau \mathrm{\in }S\mathit{}\mathrm{(}p\mathrm{+}q\mathrm{)}$ is a $\mathrm{(}p\mathrm{,}q\mathrm{)}$ shuffle if

$\mathrm{\cdots }$

$\mathrm{\cdots }$

The set of all $\mathrm{(}p\mathrm{,}q\mathrm{)}$ shuffles is denoted by $S\mathit{}\mathrm{(}p\mathrm{,}q\mathrm{)}$.

It is clear that $S(p,q)\subset S(p+q)$. Since a $(p,q)$ shuffle iscompletely determined by how the $p$ first elements are mapped, thecardinality of $S(p,q)$ is $\left(\genfrac{}{}{0pt}{}{p+q}{p}\right)$.The wedge product of a $p$-form and a $q$-form can be defined as a sumover $(p,q)$ shuffles.