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This paper deals with distinct computational methods to enumerate the set PLR(r,s,n;m) of r × 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 ≤ 13, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method forarXiv:1908.10610v1 fatcat:hkvg24wbknci3mvcuhfbavqtiq