Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

Abstract

This paper provides an in‐depth analysis of how computer algebra systems and CSP solvers can be used to deal with the problem of enumerating and distributing the set of r×s partial Latin rectangles based on n symbols according to their weight, shape, type, or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all r,s,n≤6. As a by‐product, explicit formulas are determined for the number of partial Latin rectangles of weight up to 6. Further, to illustrate the effectiveness of the computational method, we focus on the enumeration of 3 subsets: (1) noncompressible and regular, (2) totally symmetric, and (3) totally conjugate orthogonal partial Latin squares. In particular, the former enables us to enumerate the set of seminets of point rank up to 8 and to prove the existence of 2 new configurations of point rank 8. Finally, as an illustrative application, it is also exposed a method to construct totally symmetric partial Latin squares that gives rise, under certain conditions, to new families of Lie partial quasigroup rings.

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

“…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 ].…”

“…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.…”

Computational analysis of LED circuits based on partial quasi‐group rings

“…If there are not empty cells, then this constitutes a Latin square of order n. In such a case, the pair (Q, ·) is, in turn, a quasi-group. [43][44][45] The distribution of Latin squares into isomorphism classes is known for order n ≤ 11, [37][38][39] , and that of partial Latin squares has been explicitly computed for order n ≤ 6.…”

“…Different types of partial quasi-group rings have been studied from a computational point of view. [43][44][45]…”

A Historical Perspective of the Theory of Isotopisms

“…Currently, the distribution into isotopism, isomorphism and paratopism classes of partial Latin rectangles for which 1 ≤ r, s, n ≤ 6 is known [156][157][158][159] . The number of paratopism classes of partial Latin rectangles with at most 12 non-empty cells is also known [160,161].…”