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

Abstract

The current paper deals with the enumeration and classification of the set SOR r,n of self-orthogonal r × 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 Gröbner 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 × s partial Latin rectangles based on n symbols according to their size is also obtained, for all r, s, n ≤ 4.

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

“…Particularly, the presence of the monomial x i k x i ′ k as generator of the ideal I r,s,n involves the nonexistence of the symbol k twice in the j th column; that of x i k x i ′ k involves the nonexistence of the symbol k twice in the i th row; and that of x i k x i k ′ involves the nonexistence of 2 distinct symbols in the cell (i, j). On the basis of this result, the specialized algorithm described by Dickenstein and Tobis 47 was implemented in Falcón 8 for computing the cardinality of  r,s,n;m , for all r, s, n ≤ 4. For higher orders, however, the required computational cost turned out to be excessive because of large memory storage requirements.…”

“…To test its efficiency, we have firstly checked the known cardinality of  r,s,n;m , for all r, s, n ≤ 4 (see Table 2), which was already computed in Falcón. 8 In the same computer system, an Intel Core i7-2600 CPU (8 cores), with a 3.4 GHz processor and 16 GB of RAM, the maximum running time decreases from 50 seconds in Falcón 8 to less than 1 second. This corresponds to the computation of the series | 4,4,4;m |.…”

“…x s i x s kl x t i q x t klq = 0, for all i, , k, l, , q ≤ n; s, t ≤ 3; such that (i, ) ≠ (k, l), s ≤ t. d) The set TCO n;m is identified with the set of zeros of (5), (7), and (8).…”

“…[30][31][32][33] Further, the orthogonality among conjugates of a partial Latin square was indirectly contemplated [34][35][36] by focusing on the existence of incomplete Latin squares that are orthogonal to one of their conjugates and have an empty subsquare that can be filled by means of a Latin square that is orthogonal in turn to its corresponding conjugate. A more general case was proposed by the first author, 8 who makes use of computational algebraic geometry to enumerate the set of self-orthogonal partial Latin squares of order n ≤ 4. This paper delves into this topic by dealing with the sets of partial Latin squares of a given order for which their 6 conjugates either coincide or are all of them distinct and pairwise orthogonal, respectively.…”

Santilli Autotopisms of Partial Groups

“…In Section 3, a classification of the Santilli autotopisms of groups of order n ≤ 6 is explicitly given. Remark that, even if the number of quasigroups is known for order up to 11 [16,17], that of partial quasigroups is only known for order up to four [18,19].…”