  An inverted semistandard Young tableau is a row-standard tableau along with a
collection of inversion pairs that quantify how far the tableau is from being
column semistandard. Such a tableau with precisely $k$ inversion pairs is said
to be a $k$-inverted semistandard Young tableau. Building upon earlier work by
Fresse and the author, this paper develops generating functions for the numbers
of $k$-inverted semistandard Young tableau of various shapes $\lambda$ and
contents $\mu$. An easily-calculable generating function is given for the
number of $k$-inverted semistandard Young tableau that "standardize" to a fixed
semistandard Young tableau. For $m$-row shapes $\lambda$ and standard content
$\mu$, the total number of $k$-inverted standard Young tableau of shape
$\lambda$ are then enumerated by relating such tableaux to $m$-dimensional
generalizations of Dyck paths and counting the numbers of "returns to ground"
in those paths. In the rectangular specialization of $\lambda = n^m$ this
yields a generating function that involves $m$-dimensional analogues of the
famed Ballot numbers. Our various results are then used to directly enumerate
all $k$-inverted semistandard Young tableaux with arbitrary content and two-row
shape $\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux
with two-column shape $\lambda=2^n$.
