Permutations of letters under some conditions

Question

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 $p\ge 2$?

Remark : This question has been asked previously on math.SE without receiving any answers.

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?

Answer 1

Let $W_P(p,q,r,s)$ be the number of such words starting by $P$ and ending by $P$, and let $$ f_P(x,y,z,t)=\sum_{p,q,r,s}W_P(p,q,r,s)x^py^qz^rt^s. $$ Define $f_Q$, $f_R$, $f_S$ similarly (all words start by $P$!). Then $$ \pmatrix{f_P\\f_Q\\f_R\\f_S} = \pmatrix{0&x&x&x\\y&0&y&y\\z&z&0&z\\t&t&t&0}\pmatrix{f_P\\f_Q\\f_R\\f_S} +\pmatrix{x\\0\\0\\0}, $$ which implies $$ \pmatrix{f_P\\f_Q\\f_R\\f_S} =\pmatrix{1&-x&-x&-x\\-y&1&-y&-y\\-z&-z&1&-z\\-t&-t&-t&1}^{-1}\pmatrix{x\\0\\0\\0}. $$ Thus the generating function for your $W(p,q,r,s)$ is $$ f(x,y,z,t)=\pmatrix{0&1&1&1}\pmatrix{1&-x&-x&-x\\-y&1&-y&-y\\-z&-z&1&-z\\-t&-t&-t&1}^{-1}\pmatrix{x\\0\\0\\0}, $$ which, accorging to Maple, is $$ f(x,y,z,t)=\frac{x(y+z+t+2yz+2yt+2zt+3yzt)}{1-xy-xz-xt-yz-yt-zt-2xyz-2xyt-2xzt-2yzt-3xyzt}. $$ I doubt there is a much closer form. But you may find the expressions like yours in particular cases from here.