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 ${\mathcal{S}\mathcal{O}\mathcal{R}}_{r,n}$ of self-orthogonal $r\times r$ 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 ${\mathcal{S}\mathcal{O}\mathcal{R}}_{r,n}$. In particular, the cardinality of this set is shown for $r\le 4$ and $n\le 9$ and several formulas on the cardinality of ${\mathcal{S}\mathcal{O}\mathcal{R}}_{r,n}$ are exposed, for $r\le 3$. The distribution of $r\times s$ partial Latin rectangles based on $n$ symbols according to their size is also obtained, for all $r,s,n\le 4$.

15 pages, 1 figure, 4 tables. Accepted for publication in European Journal of Combinatorics

References in corpus (1)

• The set of autotopisms of partial Latin squares

Cited by in corpus (1)

• A computational algebraic geometry approach to classify partial Latin rectangles