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

Raúl M. Falcón, Óscar J. Falcón, Juan Núñez
2018 Mathematical methods in the applied sciences  
This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying r× s partial Latin rectangles based on n symbols of a given size, 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 size up to six. We focus then on the
more » ... study of non-compressible regular partial Latin squares and their equivalent incidence structure called seminet, whose distribution into main classes is explicitly determined for point rank up to eight. We prove in particular the existence of two new configurations of point rank eight.
doi:10.1002/mma.4820 fatcat:7gaagajuzbf3hkhmwcg6j6yvoe