$(p,q)$ 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 $\left(\genfrac{}{}{0pt}{}{p+q}{q}\right)$.