# $(p,q)$ unshuffle

Let $p$ and $q$ be positive natural numbers. Further, let $S(k)$ be the symmetric group^{} on the numbers $\{1,\mathrm{\dots},k\}$. A permutation^{} $\tau \in S(p+q)$ is a $(p,q)$ *unshuffle* if there exist $$ and $$ s.t.

$$\tau ({i}_{1})=1,\mathrm{\dots},\tau ({i}_{p})=p$$ |

and

$$\tau ({j}_{1})=p+1\mathrm{\dots},\tau ({j}_{q})=p+q.$$ |

Alternatively a $(p,q)$ *unshuffle* is a permutation $\tau \in S(p+q)$ s.t. ${\tau}^{-1}$ is a $(p,q)$ shuffle.

Since a $(p,q)$ unshuffle is completely determined by $\{{i}_{1},\mathrm{\dots},{i}_{p}\}$, the cardinality of $\{\sigma \in S(p+q)|\sigma \text{is an unshuffle}\}$ is $\left(\genfrac{}{}{0pt}{}{p+q}{q}\right)$.

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

Canonical name | pqUnshuffle |

Date of creation | 2013-03-22 16:47:45 |

Last modified on | 2013-03-22 16:47:45 |

Owner | Karid (16341) |

Last modified by | Karid (16341) |

Numerical id | 10 |

Author | Karid (16341) |

Entry type | Definition |

Classification | msc 20B99 |

Classification | msc 05A05 |