# $(p,q)$ shuffle

###### Definition.

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

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

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.

Title $(p,q)$ shuffle pqShuffle 2013-03-22 13:33:59 2013-03-22 13:33:59 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 20B99 msc 05A05 shuffle ShuffleOfLanguages