  Quasi-Yamanouchi tableaux connect the two most studied types of tableaux.
They are a subset of semistandard Young tableaux that are also a refinement on
standard Young tableaux, and they can be used to improve the fundamental
quasisymmetric expansion of Schur polynomials. We prove a product formula for
enumerating certain quasi-Yamanouchi tableaux and provide strong evidence that
no product formula exists in general for other shapes. Along the way, we also
prove some nice properties of their distribution and symmetry.
