Enumerating Partial Latin Rectangles

Abstract

This paper deals with different 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 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 $m \leqslant 13$, 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 $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

“…It follows similarly to the approach concerning the equivalence relation of being isotopic, which was described in (Theorem 13, [18]). To this end, for each positive integer n, let us consider the set of 3n 2 variables…”

“…Isotopic and isomorphic are equivalence relations among partial Latin squares. The distribution into such classes is known for order n ≤ 11 in the case of dealing with Latin squares [9][10][11] and for order n ≤ 6 in the case of dealing with partial Latin squares [17,18]. Partial transpose are partial isotopic are two other binary relations among partial Latin squares of the same order and weight, whose study is still in teh very initial stage.…”

“…Then, for each positive integer m ≤ n 2 , it is known [18] (see also [27,29] for a pair of first approaches in this regard) that the set L n;m is uniquely identified with the set of zeros of the affine algebraic set of the following ideal in Q[X PT n ].…”

“…Distributing (partial) Latin squares into isomorphism classes indeed constitutes a main problem in the theory of (partial) Latin squares. Currently, only the number of isomorphism classes of Latin squares of order n ≤ 11 is known [9][10][11], as well as that of partial Latin squares of order n ≤ 6 [17,18]. To obtain these last values, the computation of reduced Gröbner bases of ideals associated to partial Latin squares has played a relevant role.…”

“…Such a computation is, however, extremely sensitive to the number of involved variables and the length and degree of the corresponding generators [19][20][21][22]. Thus, although distinct techniques concerning computational algebraic geometry have been implemented since the original work of Bayer [23] for solving the classical problems of counting, enumerating and classifying partial Latin squares [17,18,[24][25][26][27][28][29] and solving related problems such as completing sudokus [30][31][32], their computational cost makes it very difficult to deal with partial Latin squares of high orders.…”