  Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and
refine standard Young tableaux. They are closely tied to the descent set of
standard Young tableaux and were introduced by Assaf and Searles to tighten
Gessel's fundamental quasisymmetric expansion of Schur functions. The descent
set and descent statistic of standard Young tableaux repeatedly prove
themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make
appearances in many ways outside of their original purpose. Some examples,
which we present in this paper, include the Schur expansion of Jack
polynomials, the decomposition of Foulkes characters, and the bigraded
Frobenius image of the coinvariant algebra. While it would be nice to have a
product formula enumeration of quasi-Yamanouchi tableaux in the way that
semistandard and standard Young tableaux do, it has previously been shown by
the author that there is little hope on that front. The goal of this paper is
to address a handful of the numerous alternative enumerative approaches. In
particular, we present enumerations of quasi-Yamanouchi tableaux using $q$-hit
numbers, semistandard Young tableaux, weighted lattice paths, and symmetric
polynomials, as well as the fundamental quasisymmetric and monomial
quasisymmetric expansions of their Schur generating function.
