# $(p,q)$ 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 is completely determined by how the $p$ first elements are mapped, the cardinality 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 sum over $(p,q)$ shuffles.

Title | $(p,q)$ shuffle |
---|---|

Canonical name | pqShuffle |

Date of creation | 2013-03-22 13:33:59 |

Last modified on | 2013-03-22 13:33:59 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 20B99 |

Classification | msc 05A05 |

Synonym | shuffle |

Related topic | ShuffleOfLanguages |