  This paper deals with distinct computational methods to enumerate the set
$\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols
with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size
of this set is a symmetric polynomial of degree $3m$, and we determine the
leading terms (the monomials of degree $3m$ through $3m-9$) using
inclusion-exclusion. For $m \leq 13$, exact formulas for these symmetric
polynomials are determined using a chromatic polynomial method. Adapting Sade's
method for enumerating Latin squares, we compute the exact size of
$\mathrm{PLR}(r,s,n;m)$, for all $r \leq s \leq n \leq 7$, and all $r \leq s
\leq 6$ when $n=8$. Using an algebraic geometry method together with Burnside's
Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leq s
\leq n \leq 6$. Numerical results have been cross-checked where possible.
