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.