Let $W(p,q,r,s)$ be the number of permutations of the letters which satisfy the following conditions :
Condition 1 : The letters are consist of $P,Q,R,S$.
Condition 2 : The number of letter $P,Q,R,S$ is $p,q,r,s$ respectively.
Condition 3 : Any two adjacent letters are different from each other.
Condition 4 : The first letter is $P$, and the last letter is not $P$.
Then, here is my question.
Question : Can we get a closed-form expression of $W(p,q,r,s)$ for $p\ge 2$?
Motivation : I've just got the following closed-form expression of $W(1,q,r,s)$ :
$W(1,q,r,s)=$ $$\sum_{k=1}^{q+1}\binom{r-1}{k-1}\left\{\binom{q-1}{k}\binom{2k}{q-1+r-s}+2\binom{q}{k}\binom{2k}{q+r-s}+\binom{q+1}{k}\binom{2k}{q+1+r-s}\right\}$$
However, I'm facing difficulty for the $p\ge 2$ cases. Can anyone help?
Update : I crossposted to MO.
