Factorization of certain Macdonald Littlewood-Richardson coefficients

We find and prove a factorization formula for certain Macdonald Littlewood-Richardson coefficients $c_{\lambda \mu }^{\nu }(q,t)$. Namely, we consider the case that the Kostka number $K_{\mu ,\nu -\lambda }$ is $1$. This settles a particular case of a more general conjecture of Richard Stanely. This conjecture proposes that a factorization formula exists whenever the corresponding regular Littlewood-Richardson coefficient $c_{\lambda \mu }^{\nu }$ is $1$.