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

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 $r\times s$ 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 $r,s,n\leq 6$. As a by-product, explicit formulas are determined for the number of partial Latin rectangles of weight up to six. Further, in order to illustrate the effectiveness of the computational method, we focus on the enumeration of three subsets: (a) non-compressible and regular, (b) totally symmetric, and (c) totally conjugate orthogonal partial Latin squares. In particular, the former enables us to enumerate the set of seminets of point rank up to eight and to prove the existence of two new configurations of point rank eight. 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