# $(p,q)$ unshuffle

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

 $\tau(i_{1})=1,\ldots,\tau(i_{p})=p$

and

 $\tau(j_{1})=p+1\ldots,\tau(j_{q})=p+q.$

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 $\{i_{1},\ldots,i_{p}\}$, the cardinality of $\{\sigma\in S(p+q)|\mbox{\sigma is an unshuffle}\}$ is $\binom{p+q}{q}$.

Title $(p,q)$ unshuffle pqUnshuffle 2013-03-22 16:47:45 2013-03-22 16:47:45 Karid (16341) Karid (16341) 10 Karid (16341) Definition msc 20B99 msc 05A05