Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

Abstract

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 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.

Covers and partial transversals of Latin squares

“…For any (n + 1)-cover of a Latin square L, the corresponding n + 1 vertices of the Latin square graph Γ L induce a subgraph with 3 edges. (This is an example of a partial Latin square graph [10]. A partial Latin square is a matrix in which entries are either empty or contain a single symbol, and no symbol is repeated within any row or column.…”

A Historical Perspective of the Theory of Isotopisms

“…Thus, for instance, if an isotopism (respectively, isomorphism or paratopism) preserves a given partial Latin rectangle, the former is said to be an autotopism (respectively, automorphism or autoparatopism) of the latter. Currently, the enumeration of autotopisms of a given partial Latin rectangle is an open problem that requires the description of different isotopism invariants [146,148,162,163].…”

“…A current open problem to be solved here is the study of totally symmetric partial Latin squares having a given autotopism group. In [163], it is considered the more general problem in which the autoparatopism group is any subgroup of the symmetric group S 3 .…”

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

“…[8][9][10][11] The study of combinatorial designs by means of electric circuits was already considered by Brooks et al 12 and Tjur 13 . More specifically, we focus on the synthesis and computational analysis of light-emitting diode (LED) circuits through which the flow of electric current is regulated by a set of devices (voltage regulators, single-pole double-throw (SPDT) relays, analog multipliers, capacitors, resistors, push buttons, and, of course, LEDs) whose topology and working are uniquely described by a given partial quasi-group ring, or, equivalently, by a partial Latin square.…”

“…Currently, the search of such invariants constitutes an open and active problem. [8][9][10][11] The study of combinatorial designs by means of electric circuits was already considered by Brooks et al 12 and Tjur 13 . Further, Kumar et al 14 dealt with the completion problem of partial Latin squares motivated by its possible application to light path assignments and switch configurations.…”