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 SORr,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 SORr,n. In particular, the cardinality of this set is shown for r ≤ 4 and n ≤ 9 and several formulas on the cardinality of SORr,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.