A $(p,q)$-shuffle is a permutation $\sigma$ of the set $\{1, \dots, p, p+1, \dots,p+q\}$ such that $$\sigma(1)<\dots<\sigma(p)$$ and $$\sigma(p+1) < \dots<\sigma(p+q)\,.$$

It is known that the number of $(p,q)$-shuffles is ${p+q \choose p}$.

Looking at the case $p+q = 6$ and considering the parity of shuffles, I was surprised to find that

- $3$ of the ${6 \choose 1} = 6$ $(1,5)$-shuffles have even parity
- $9$ of the ${6 \choose 2} = 15$ $(2,4)$-shuffles have even parity
- $10$ of the ${6 \choose 3} = 20$ $(3,3)$-shuffles have even parity
- $9$ of the ${6 \choose 4} = 15$ $(4,2)$-shuffles have even parity
- $3$ of the ${6 \choose 5} = 6$ $(5,1)$-shuffles have even parity

It's not so difficult to believe that the number of even $(p,q)$-shuffles should equal the number of even $(q,p)$-shuffles. But I was very surprised to see that there are so many more even than odd $(2,4)$-shuffles.

Are any closed-form solutions known for the number of even $(p,q)$-shuffles? If not, is anything known about the asymptotic behaviour?