Falcón Ganfornina, Raúl ManuelStones, Rebecca J.2024-10-232024-10-232020Falcón Ganfornina, R.M. y Stones, R.J. (2020). Enumerating partial Latin rectangles. Electronic Journal of Combinatorics, 27 (2). https://doi.org/10.37236/9093.1077-8926https://hdl.handle.net/11441/163987This paper deals with different 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 given by 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 6 13, exact formu las 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 6 s 6 n 6 7, and all r 6 s 6 6 when n = 8. Using an algebraic geometry method together with Burnside’s Lemma, we enumer ate isomorphism, isotopism, and main classes when r 6 s 6 n 6 6. Numerical results have been cross-checked where possible.application/pdf41 p.engAtribución-NoComercial 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/Partial Latin rectangleIsomorphismIsotopismMain classInclusion -exclusionChromatic polynomialAlgebraic geometryinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.37236/9093