Definitions

# (p,q) shuffle

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

$tau\left(1\right) < cdots < tau\left(p\right) ,$,
$tau\left(p+1\right) < cdots < tau\left(p+q\right) ,$.

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

It is clear that

$S\left(p,q\right)subset S\left(p+q\right).$

Since a $\left(p,q\right)$ shuffle is completely determined by how the $p$ first elements are mapped, the cardinality of $S\left(p,q\right)$ is

$\left\{p+q choose p\right\}.$

The wedge product of a $p$-form and a $q$-form can be defined as a sum over $\left(p,q\right)$ shuffles.

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