(p,q) shuffle

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

$\tau (1)<\cdots <\tau (p)\,$,

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

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.

This article incorporates material from (p,q) shuffle on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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