Raúl M. Falcón

The set of autotopisms of partial Latin squares

Abstract Symmetries of a partial Latin square are primarily determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism Θ , the cardinality of the set PLS Θ of partial Latin squares which are invariant under Θ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties were studied. It is also seen that the cycle structure of Θ determines the possible sizes of the elements of PLS Θ and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square.

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

The current paper deals with the enumeration and classification of the set SOR r , n of self-orthogonal r i� r partial Latin rectangles based on n symbols. These combinatorial objects are identified with the independent sets of a Hamming graph and with the zeros of a radical zero-dimensional ideal of polynomials, whose reduced Grobner basis and Hilbert series can be computed to determine explicitly the set SOR r , n . In particular, the cardinality of this set is shown for r � 4 and n � 9 and several formulas on the cardinality of SOR r , n are exposed, for r � 3 . The distribution of r i� s partial Latin rectangles based on n symbols according to their size is also obtained, for all r , s , n � 4 .

Classifying partial Latin rectangles

Abstract Isotopisms of the set R r , s , n of r × s partial Latin rectangles based on n symbols constitute a finite group that acts on this set by permuting rows, columns and symbols. The number of partial Latin rectangles preserved by this action only depends on the conjugacy classes of these permutations. In this paper, the distribution of the isotopism group into conjugacy classes is considered in order to determine the distribution of R r , s , n into isomorphism and isotopism classes, for all r , s , n ≤ 6 .

Study of Critical Sets in Latin Squares by using the Autotopism Group

Abstract Given a Latin square L and a subset F of its autotopism group U ( L ) , we study in this paper some properties and results which partial Latin squares contained in L inherit from U ( L ) , by using F . In particular, we define the concept of F -critical set of L and we ask ourselves about the smallest one contained in L.

Counting and enumerating feasible rotating schedules by means of Gröbner bases

This paper deals with the problem of designing and analyzing rotating schedules with an algebraic computational approach. Specifically, we determine a set of Boolean polynomials whose zeros can be uniquely identified with the set of rotating schedules related to a given workload matrix subject to standard constraints. These polynomials constitute zero-dimensional radical ideals, whose reduced Grobner bases can be computed to count and even enumerate the set of rotating schedules that satisfy the desired set of constraints. Thereby, it enables to analyze the influence of each constraint in the same.

Partial Latin Squares Having a Santilli’s Autotopism in their Autotopism Groups

Abstract An autotopism Θ = (α, β, γ) of a partial Latin square P = (pi, j ) with symbols in N = {0, 1, …, n − 1} is said to be a Santilli’s autotopism of P if there exist jalpha;, jβ,jγ ∈ N such that α(i) = pi, jα , β(i) = pi, jβ and γ(i) = pi, jγ , for all i ∈ N. In this paper, we are interested in the set SPLS (Θ) of partial Latin squares having Θ as a Santilli’s autotopism. Specifically, the cardinality of the set of minimal elements of SPLS(Θ) is obtained and a classification of Latin squares Santilli’s isotopisms of order ≤ 5 fixing at least one symbol is given.

Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field

Abstract Evolution algebras were introduced into Genetics to deal with the mechanism of inheritance of asexual organisms. Their distribution into isotopism classes is uniquely related with the mutation of alleles in non-Mendelian Genetics. This paper deals with such a distribution by means of Computational Algebraic Geometry. We focus in particular on the two-dimensional case, which is related to the asexual reproduction processes of diploid organisms. Specifically, we determine the existence of four isotopism classes, whatever the base field is, and we characterize the corresponding isomorphism classes.

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

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.

Autoparatopism stabilized colouring games on rook's graphs

Abstract We introduce the autoparatopism variant of the autotopism stabilized colouring game on the n × n rooks graph as a natural generalization of the latter so that each board configuration is uniquely related to a partial Latin square of order n that respects a given autoparatopism (θ; π). To this end, we distinguish between π ∈ { Id , ( 12 ) } and π ∈ { ( 13 ) , ( 23 ) , ( 123 ) , ( 132 ) } . The complexity of this variant is examined by means of the autoparatopism stabilized game chromatic number. Some illustrative examples and results are shown.

Two-line graphs of partial Latin rectangles

Abstract Two-line graphs of a given partial Latin rectangle are introduced as vertex-and-edge-coloured bipartite graphs that give rise to new autotopism invariants. They reduce the complexity of any currently known method for computing autotopism groups of partial Latin rectangles.