Enumerating partial Latin rectangles

Abstract

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 for enumerating Latin squares, we compute the exact size of PLR(r, s, n; m), for all r ≤ s ≤ n ≤ 7, and all r ≤ s ≤ 6 when n = 8. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when r ≤ s ≤ n ≤ 6. Numerical results have been cross-checked where possible.